Friday, January 5, 2024

The Seminars on Undergraduate Mathematics in Montreal (SUMM) is organized by undergraduate students from Montreal universities. The main objective is to create an environment facilitating exchange of ideas and interests as well as allowing students to network.

The SUMM weekend is aimed at undergraduate students in mathematics or related domains. This year, the conference will be held from **January 5-7, 2024**.

The weekend consist of two days of presentations given by undergraduate students and invited professors. The presentations can cover a broad range of subjects, from mathematical physics to the applications of artificial intelligence as well as the history and philosophy of mathematics.

During the SUMM, students can choose to give a talk, or simply to attend presentations given by their peers. It's an occasion to share the passion for mathematics in a stimulating environment, while networking with other passionate students over the weekend.

We hope to see you there!

Friday, December 8, 2023

**Computational Complexity in Algebraic Combinatorics**

Algebraic Combinatorics studies objects and quantities originating in Algebra, Representation Theory and Algebraic Geometry via combinatorial methods, finding formulas and neat interpretations. Some of its feats include the hook-length formula for the dimension of an irreducible symmetric group ($S_n$) module, or the Littlewood-Richardson rule to determine multiplicities of GL irreducibles in tensor products. Yet some natural multiplicities elude us, among them the Kronecker coefficients for the decomposition of tensor products of $S_n$ irreducibles, and the plethysm coefficients for compositions of GL modules. Answering those questions could help Geometric Complexity Theory establish lower bounds for the far reaching goal to show that $P \neq NP$.

We will discuss how Computational Complexity Theory provides a theoretical framework for understanding what kind of formulas or rules we could actually have. We will use this to show that the square of a symmetric group character could not have a combinatorial interpretation.

Based on joint works with Christian Ikenmeyer and Igor Pak.

**Date: **December 8, 2023, 3:30 PM

**Place: **UQAM, President-Kennedy Building, 201, ave du Président-Kennedy, room PK-5115

Friday, November 24, 2023

**State-dependent Sampling in Observational Cohort Studies**

Observational cohort studies of chronic disease involve the recruitment and follow-up of a sample of individuals with the goal of learning about the course of the disease, the effect of fixed and time-varying risk factors. Analysis of this information is often facilitated by using multistate models with intensity functions governing transition between disease states. Chronic disease studies often involve conditions for recruitment, for example incident cohort involves individuals who are healthy at accrual, prevalent cohort samples individuals who have already developed the disease, and a length biased sampling includes individual who are alive at the time of recruitment. In this talk we discuss the impact of ignoring state-dependent sampling in life history analysis and the ways of addressing the issue using auxiliary information. A longitudinal study of aging and cognition among religious sisters is used to illustrate the related methodology.

**Date: **November 24, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Friday, November 17, 2023

**The local Langlands program and characteristic p**

Let *n* be an integer greater than or equal to 2. The Langlands program for GLn connects *n*-dimensional representations of Galois groups and infinite dimensional representations of GLn. For the purposes of number theory, we are led to consider all these representations on vector spaces over fields of characteristic *p* for *p* an arbitrary prime number. On the GLn side, this leads in particular to the following problem: understand and construct the - or some - representations of GLn(K) in characteristic p where K is a finite extension of the field of p-adic numbers Qp (for the same prime number *p*!), and most importantly those of these representations which appear on cohomology spaces. This problem has challenged experts for more than 20 years and is still largely open even for GL2(K). I will recall the history, the difficulties encountered, and will state some recent results for GL2(K).

**Date: **November 17, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Friday, November 10, 2023

**Model Based optimization with convex-composite optimization**

Optimization problems have an enormous variety and complexity and solving them requires techniques for exploiting their underlying mathematical structure. The modeler needs to balance model complexity with computational tractability as well as viable techniques for post optimal analysis and stability measures.

In this talk we describe the convex-composite modeling framework which covers a broad range of optimization problems including nonlinear programming, feasibility, minimax optimization, sparcity optimization, feature selection, Kalman smoothing, parameter selection, and nonlinear maximum likelihood to name a few. The goal is to identify and exploit the underlying convexity that a given problem may possess since convexity allows one to tap into the very rich theoretical foundation as well as the wide range of highly efficient numerical methods available for convex problems. The systematic study of convex-composite problems began in the 1970's concurrent with the emergence of modern nonsmooth variational analysis. The synergy between these ideas was natural since convex-composite functions are neither convex nor smooth. The recent resurgence in interest for this problem class is due to emerging methods for approximation, regularization and smoothing as well as the relevance to a number of problems in global health, environmental modeling, image segmentation, dynamical systems, signal processing, machine learning, and AI. In this talk we review the convex-composite problem structure and variational properties. We then discuss algorithm design and if time permits, we discuss applications to filtering methods for signal processing.

**Date: **November 10, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Friday, November 3, 2023

**Point Counting over finite fields and the cohomology of moduli spaces of curves**

Algebraic geometry studies solution sets of polynomial equations. For instance, over the complex numbers, one may examine the topology of the solution set, whereas over a finite field, one may count its points. For polynomials with integer coefficients, these two fundamental invariants are intimately related via cohomological comparison theorems and trace formulas for the action of Frobenius. I will present recent results regarding point counts over finite fields and the cohomology of moduli spaces of curves that resolve longstanding questions in algebraic geometry and confirm more recent predictions from the Langlands program.

**Date: **November 3, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Friday, October 27, 2023

**(Conférence Chaire Aisenstadt) On the hunt for chirps**

The detection of specific oscillatory behaviours in signals is a key issue in turbulence, gravitational waves, or physiological data, where they turn out to be the signature of important phenomena localized in time or space. We will show how some recently introduced methods of harmonic analysis allow us to characterize and classify such behaviors, and ultimately yield numerical methods to perform their detection.

**Date: **October 27, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 6214

Friday, October 20, 2023

**Billiards in conics revisted **

Optical properties of conics have been known since the classical antiquity. The reflection in an ideal mirror is also known as the billiard reflection and, in modern terms, the billiard inside an ellipse is completely integrable. The interior of an ellipse is foliated by confocal ellipses that are its caustics: a ray of light tangent to a caustic remains tangent to it after reflection (“caustic” means burning).

I shall explain these classic results and some of their geometric consequences, including the Ivory lemma asserting that the diagonals of a curvilinear quadrilateral made by arcs of confocal ellipses and hyperbolas are equal (this lemma is in the heart of Ivory's calculation of the gravitational potential of a homogeneous ellipsoid). Other applications include the Poncelet Porism, a famous theorem of projective geometry that has celebrated its bicentennial, and its lesser known ramifications, such as the Poncelet Grid theorem and the related circle patterns and configuration theorems.

**Date: **October 20, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Friday, October 13, 2023

**Patterns in tri-block co-polymers: droplets, double-bubbles and core-shells, and a "new" partitioning problem**

We study the Nakazawa-Ohta ternary inhibitory system, which describes domain morphologies in a triblock copolymer as a nonlocal isoperimetric problem for three interacting phase domains. The free energy consists of two parts: the local interface energy measures the total perimeter of the phase boundaries, while a longer-range Coulomb interaction energy reflects the connectivity of the polymer chains and promotes splitting into micro-domains. We consider global minimizers on the two-dimensional torus, in a limit in which two of the species have vanishingly small mass but the interaction strength is correspondingly large. In this limit there is splitting of the masses, and each vanishing component rescales to a minimizer of an isoperimetric problem for clusters in 2D. Depending on the relative strengths of the coefficients of the interaction terms we may see different structures for the global minimizers, ranging from a lattice of isolated simple droplets of each minority species to double-bubbles or core-shells. These results have led to a new type of partitioning problem that I will also introduce. These represent work with S. Alama, with X. Lu, and C. Wang, as well as with S. Vriend.

**Date: **October 13, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Friday, October 6, 2023

**(CRM-SSC Prize) Auto-regressive approximations to non-stationary time series, with inference and applications**

Understanding the time-varying structure of complex temporal systems is one of the main challenges of modern time series analysis. In this talk, I will demonstrate that a wide range of short-range dependent non-stationary and nonlinear time series can be well approximated globally by a white-noise-driven auto-regressive (AR) process of slowly diverging order. Uniform statistical inference of the latter AR structure will be discussed through a class of high-dimensional L2 tests. I will further discuss applications of the AR approximation theory to globally optimal short-term forecasting, efficient estimation, and resampling inference under complex temporal dynamics.

**Date: **October 6, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 6214

Wednesday, October 4, 2023

**October 4-7, 2023**

The *relative Langlands program* was born out of the methods used to study automorphic *L*-functions by representing them by various integrals of automorphic forms. The work of Jacquet, D. Prasad, and others, highlighted the connections between Langlands functoriality and the problem of *distinction* - or, harmonic analysis on certain (almost) homogeneous *G*-spaces *X*, such as spherical varieties. The conjectures of Gan-Gross-Prasad and Ichino-Ikeda, based on work of Waldspurger and many others, revealed a pattern that relates *global* integrals of automorphic forms to local harmonic analysis. The work of Gaitsgory-Nadler and Sakellaridis-Venkatesh allowed the formulation of a general program, based on the *dual group* of a spherical variety. The goal of this workshop will be to introduce the more recent work of Ben-Zvi-Sakellaridis-Venkatesh, which takes a step further, introducing a *categorical* version of the relative Langlands program.

Friday, September 29, 2023

**On strong solutions of time inhomogeneous Itô's equations with Morrey diffusion gradient and drift and related PDEs results. A supercritical case.**

We prove the existence of strong solutions of Itô's stochastic time dependent equations with irregular diffusion and drift terms of Morrey spaces. Strong uniqueness is also discussed. The results are new even if there is no drift. The results are based on the solvability of parabolic equations with Morrey

drift in Morrey spaces, which is also new.

**Date: **September 29, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Friday, September 15, 2023

**Branching processes in random matrix theory and analytic number theory**

The limiting distributions for maxima of independent random variables have been classified during the first half of last century. This classification does not extend to strong interactions, in particular to the flurry of processes with natural logarithmic (or multiscale) correlations. These include branching random walks or the 2d Gaussian free field. More recently, Fyodorov, Hiary and Keating (2012) exhibited new examples of log-correlated phenomena in number theory and random matrix theory. As a result (and as a testing ground of their observations) they have formulated very precise conjectures about maxima of the characteristic polynomial of random matrices, and the maximum of L-functions on typical interval the critical line. I will describe the recent progress towards these conjectures in both the random and deterministic setting.

**Date: **September 15, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Monday, June 12, 2023

**June 12-16, 2023**

The study of lengths of closed geodesics and of the Laplace spectrum on hyperbolic surfaces was initiated by work of Huber in the 1970’s where he used the Selberg trace formula to determine various asymptotics and bounds. Another major development came with Mirzakhani’s thesis and subsequent works, in which she developed new techniques to integrate functions over moduli spaces of hyperbolic surfaces. This opened the way for proving results about random surfaces or for proving existence results based on probabilistic arguments. In the past few years, there has been a renewed interest for the Selberg trace formula, especially in combination with Mirzakhani’s integration technique. Various other models of random surfaces have also been studied with great success, and we now understand the behaviour of several geometric invariants thanks to recent breakthroughs, but many open problems remain.

The goal of this discovery school is to introduce graduate students and advanced undergraduate students to the above circle of ideas and give them the chance to learn about recent advances from leading experts in the field.

We support the statement of inclusiveness. Everyone is welcome to attend.

Friday, June 9, 2023

The organizing committee of the XXVth Graduate Student Conference of the *Institut des Sciences Mathématiques* is pleased to welcome you to the Université de Sherbrooke in June 2023! This conference brings together the community of graduate students in mathematics from all of Quebec’s universities for a weekend each year. During the event, participants have the opportunity to attend five plenary sessions, to present their own work if they wish, and to take part in social activities.

Monday, June 5, 2023

Discrete probability explores the structure of the objects studied in discrete mathematics. In theory, the study of large random discrete structures can frequently be reduced to understanding a uniform sample from a finite set. In practice, however, for structured discrete objects (such as trees, graphs and maps), understanding what a uniform sample typically looks like frequently involves a rich interplay between combinatorial, probabilistic and algorithmic arguments. The courses in this discovery school will showcase this interplay, presenting both classical and recent results on the asymptotic behaviour of large random structures. The schedule of lectures will be relatively light, leaving plenty of time for students to discuss together to deepen their understanding of the material.

Friday, May 12, 2023

**Estimating individualized treatment rules without individual data in multicentre studies**

Estimating individualized treatment rules is challenging, as the treatment effect heterogeneity of interest often suffers from low power. This motivates the use of very large datasets such as those from multiple health systems or multicentre studies, which may raise concerns of data privacy. In this talk, I will introduce a statistical framework for of estimation individualized treatment rules and show how distributed regression can be used in combination with dynamic weighted regression to find an optimal individualized treatment rule whilst obscuring individual-level data. The robustness of this approach and its flexibility to address local treatment practices will be shown in simulation. The work is motivated by, and illustrated with, an analysis of the U.K.’s Clinical Practice Research Datalink on the treatment of depression.

**Date: **May 12, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 6214

Friday, May 5, 2023

The 2023 McGill (Bio)Statistics Research and Career Day will be held in person on Friday, May 5th. Students from McGill and other universities in Montreal are invited to give oral presentations about their recent or ongoing research projects in statistics-related fields. We will also have three keynote speakers and an exciting career panel of professionals with different types of related work experience!

Friday, May 5, 2023

**Eigenvalues and minimal surfaces**

Eigenvalues of the Laplace operator of Euclidean domains govern many physical phenomena, including heat flow and sound propagation. In particular, various inequalities for Laplace eigenvalues have fascinated mathematicians since XIXth century. The following question was first formulated by Lord Rayleigh in his “Theory of sound”: which planar domain of given area has the lowest first Dirichlet eigenvalue? This is an example of an isoperimetric eigenvalue problem for planar domains. The focus of the present talk is on more general isoperimetric problems, where one considers surfaces equipped with Riemannian metrics. More specifically, sharp upper bounds for Laplace and Steklov eigenvalues have been an active area of research for the past decade, largely due to their fascinating connection to fundamental geometric objects, minimal surfaces. We will survey recent results exploring the applications of this connection both to minimal surface theory and to isoperimetric eigenvalue problems, culminating in a surprising link between Laplace and Steklov spectra.

**Date: **May 5, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Friday, April 28, 2023

**Aggregation-Diffusion and Kinetic PDEs for collective behavior: applications in the sciences**

I will present a survey of micro, meso and macroscopic models where repulsion and attraction effects are included through pairwise potentials. I will discuss their interesting mathematical features and applications in mathematical biology and engineering. Qualitative properties of local minimizers of the interaction energies are crucial in order to understand these complex behaviors. I will showcase the breadth of possible applications with three different phenomena in applications: segregation, phase transitions and consensus.

**Date: **April 28, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Friday, April 21, 2023

**The Search for New Physics**

The standard model (SM) of particle physics explains nearly all experimental results to date. There is no doubt that it is correct. However, for a variety of reasons it is understood to be incomplete – there must be physics beyond the SM. In this talk, I provide a brief review of the SM, discuss the reasons we believe it is incomplete, and present some examples of my contributions over the years to this search for new physics.

**Date: **April 21, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 6214

Friday, March 31, 2023

**Set Theory and Topological Algebra**

We shall discuss some recent applications of set-theoretic and model-theoretic methods to the study of topological groups. In particular, we shall outline how ultra powers can be used to solve old problems of Comfort and van Douwen and introduce a new set-theoretic axiom to study convergence properties in topological groups. If time permits we may also briefly mention the use of Fraissé theory in the study of groups of homeomorphisms.

**Date: **March 31, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Friday, March 24, 2023

**Laplace Eigenfunctions and the Frequency Function Method**

A classical idea in the study of eigenfunctions of the Laplace-Beltrami operator is that they behave like polynomials of degree corresponding to the eigenvalue. In the first part of the talk we present some recent results on eigenfunctions which confirm this idea. As a corollary, we formulate a local version of the celebrated Courant theorem on the number of nodal domains of eigenfunctions. The second part of the talk is devoted to Dirichlet-Laplace eigenfunctions in subdomains of the Euclidean space. We give a sharp bound of the size of the zero set of eigenfunctions under some mild assumptions on the regularity of the boundary. Versions of the almost monotonicity of the frequency function are important tools for both parts of the talk.

**Date: **March 24, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Friday, March 17, 2023

**Sums of the Divisor Function and Randam Matric Distributions**

The divisor function gives the number of positive divisors of a natural number. How can we go about understanding the behavior of this function when going over the natural numbers? In this talk we will discuss strategies to better understand this function, issues related to the distribution of these values, and connections to the Riemann zeta function and some groups of random matrices.

**Date: **March 17, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Friday, March 10, 2023

**Some expected and unexpected applications of Riemann surface theory in mathematical physics**

Riemann surfaces and algebraic curves are ubiquitous in mathematics. Without attempting a review of the variety of ways in which they are useful, we will look at some specific examples and discuss some curious links with physics and mathematical physics, including a link between the quantum harmonic oscillator and moduli spaces of Riemann surfaces, and a link between the periodic Toda chain, Chebyshev polynomials, and Painlevé equations.

**Date: **March 10, 2023, 3:30 PM

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Wednesday, March 8, 2023

The CRM’s EDI committee warmly invites all mathematicians and statisticians who identify as women (students, postdocs or professors), and their friends, for a casual get together honouring International Women’s Day and women in science. It will be an informal occasion to (re-) connect with each other and share experiences and ideas with friends and colleagues. We are happy to have as distinguished guest Professor Vasilisa Shramchenko from Sherbrooke University who will also be giving the colloquium on Friday, March 10^{th}.

**Where:** Salon Maurice Labbé, 6^{th} Floor, Pavillon André-Aisenstadt, Université de Montréal

**Date:** March 8, 2023

**Time:** 4:00 PM – 6:00 PM

Friday, February 24, 2023

**An introduction to rigid systems**

A representation of a group G is said to be rigid, if it cannot be continuously deformed to a non-isomorphic representation. If G happens to be the fundamental group of a complex projective manifold, rigid representations are conjectured (by Carlos Simpson) to be of geometric origin. In this talk I will outline the basic properties of rigid local systems and discuss several consequences of Simpson‘s conjecture. I will then outline recent progress on these questions (joint work with Hélène Esnault) and briefly mention applications to geometry and number theory such as the recent resolution of the André-Oort conjecture by Pila-Shankar-Tsimerman.

**Date: **February 24, 2023

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Friday, February 10, 2023

**The Mathematical Foundations of Deep Learning: From Rating Impossibility to Practical Existence Theorems**

Deep learning is having a profound impact on industry and scientific research. Yet, while this paradigm continues to show impressive performance in a wide variety of applications, its mathematical foundations are far from being well established. In this talk, I will present recent developments in this area by illustrating two case studies.

First, motivated by applications in cognitive science, I will present “rating impossibility" theorems. They identify frameworks where deep learning is provably unable to generalize outside the training set for the seemingly simple task of learning identity effects, i.e. classifying whether pairs of objects are identical or not.

Second, motivated by applications in scientific computing, I will illustrate “practical existence" theorems. They combine universal approximation results for deep neural networks with compressed sensing and high-dimensional polynomial approximation theory. As a result, they yield sufficient conditions on the network architecture, the training strategy, and the number of samples able to guarantee accurate approximation of smooth functions of many variables.

Time permitting, I will also discuss work in progress and open questions.

**Date: **February 10, 2023

**Place: **Centre de recherches mathématiques, Pavillon André-Aisenstadt, Université de Montréal, Room 5340

Friday, January 27, 2023

**The distribution of Selmer groups of elliptic curves**

The Goldfeld and Katz--Sarnak conjectures predict that 50% of elliptic curves have rank 0, that 50% have rank 1, and that the average rank of elliptic curves is 1/2 (the remaining 0% of elliptic curves not interfering in the average). Successive works of Brumer, Heath-Brown, and Young, have approached this problem by studying the central values of the L functions of elliptic curves. In this talk, we will take an algebraic approach, in which we study the ranks of elliptic curves via studying their Selmer groups.

Poonen and Stoll developed a beautiful model for the behaviours of *p*-Selmer groups of elliptic curves, and gave heuristics for all moments of the sizes of these groups.

In this talk, I will describe joint work with Manjul Bhargava and Ashvin Swaminathan, in which we prove that the second moment of the size of the 2-Selmer groups of elliptic curves is bounded above by 15 (which is the constant predicted by Poonen--Stoll).

The conference will not be broadcast on Zoom.

**Date: **January 27, 2023

**Place: **McGill University, Burnside Hall, Room 1205

Friday, January 20, 2023

**Ramsey Theory, Sparsity and Limits (Combinatorics and Model Theory)**

Several combinatorial problems are treated in the context of model theory. We survey three such instances which were investigated recently, coming from Ramsey theory, sparsity of graphs and limits of sequences of structures. These are diverse areas but share some properties where the connection to model theory is non-trivial and interesting. It also presents several open problems of interest to both combinatorics and model theory.

**Date**: January 20, 2023, 3:30 pm

**Place: **Pavillon André Aisenstadt, Room 5340, 2920, chemin de la tour, Montréal

Friday, January 13, 2023

**Sticky Kakeya sets, and the sticky Kakeya conjecture**

A Kakeya set is a compact subset of R^{n} that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that such sets must have dimension n. This conjecture is closely related to several open problems in harmonic analysis, and it sits at the base of a hierarchy of increasingly difficult questions about the behavior of the Fourier transform in Euclidean space.

There is a special class of Kakeya sets, called sticky Kakeya sets. Sticky Kakeya sets exhibit an approximate self-similarity at many scales, and sets of this type played an important role in Katz, Łaba, and Tao's groundbreaking 1999 work on the Kakeya problem. In this talk, I will discuss a special case of the Kakeya conjecture, which asserts that sticky Kakeya sets must have dimension n. I will discuss the proof of this conjecture in dimension 3. This is joint work with Hong Wang.

**Date**: January 13, 2023, 3:30 pm

**Place: **Pavillon André Aisenstadt, Room 5340, 2920, chemin de la tour, Montréal

Friday, January 6, 2023

The Seminars on Undergraduate Mathematics in Montreal (SUMM) is organized by undergraduate students from Montreal universities. The main objective is to create an environment facilitating exchange of ideas and interests as well as allowing students to network.

The SUMM weekend is aimed at undergraduate students in mathematics or related domains. This year, the conference will be held from January 6 to January 8, 2023.

The weekend consist of two days of presentations given by undergraduate students and invited professors. The presentations can cover a broad range of subjects, from mathematical physics to the applications of artificial intelligence as well as the history and philosophy of mathematics.

During the SUMM, students can choose to give a talk, or simply to attend presentations given by their peers. It's an occasion to share the passion for mathematics in a stimulating environment, while networking with other passionate students over the weekend.

We hope to see you there!

Friday, December 9, 2022

**Ergodic theory of the stochastic Burgers equation**

I am interested in stationary distributions for the Burgers equation with random forcing. I will first consider an oversimplified random dynamical system to illustrate the power of a general approach based on the so-called pullback procedure. For the Burgers equation, which is a basic evolutionary stochastic PDE of Hamilton-Jacobi type related to fluid dynamics, growth models, and the KPZ equation, one can realize this approach via studying long-term properties of random Lagrangian action minimizers and directed polymer measures in random environments. The compact space case was studied in 2000's. This talk is based on my work on the noncompact case, joint with Eric Cator, Kostya Khanin, Liying Li.

**Date**: December 9, 2022, 3:30 pm

**Place: **Pavillon André Aisenstadt, Room 5340, 2920, chemin de la tour, Montréal

Friday, December 2, 2022

**p-adic methods for solving Diophantine equations**

The problem of finding rational or integer solutions to polynomial equations is one of the oldest problems in mathematics and is one of the key driving forces in the development of Number Theory. In the last 15 years new methods were developed that can sometimes effectively solve this problem. These methods attempt to find the solutions inside the larger set of solutions of the same equation in the field of p-adic numbers as the vanishing set of some computable function. When these methods work they give the rational solutions to arbitrarily large p-adic precision, which usually suffices to rigorously recover the full set of solutions.

I will survey the new methods, originating from the work of Kim and from the more recent work of Lawrence and Venkatesh. I will then explain my work with Muller and Srinivasan that uses a p-adic version of the notion of norms on line bundles and associated heights, as used for example in arithmetic dynamics, to give a new approach to some Kim type results.

**Date**: December 2, 2022, 3:30 pm

**Place: **Concordia University, Library Building, 9th floor, room LB 921-4

Friday, November 25, 2022

**(Almost) all of Entity Resolution**

Whether the goal is to estimate the number of people that live in a congressional district, to estimate the number of individuals that have died in an armed conflict, or to disambiguate individual authors using bibliographic data, all these applications have a common theme — integrating information from multiple sources. Before such questions can be answered, databases must be cleaned and integrated in a systematic and accurate way, commonly known as record linkage, de-duplication, or entity resolution. In this article, we review motivational applications and seminal papers that have led to the growth of this area. Specifically, we review the foundational work that began in the 1940’s and 50’s that have led to modern probabilistic record linkage. We review clustering approaches to entity resolution, semi- and fully supervised methods, and canonicalization, which are being used throughout industry and academia in applications such as human rights, official statistics, medicine, citation networks, among others. Finally, we discuss current research topics of practical importance. This is joint work with Olivier Binette.

**Date: **November 25, 2022

**To obtain the zoom link:** https://forms.gle/8cqLPYuXwiqdVU1L8

Friday, November 18, 2022

**Quantitative Stability in the Calculus of Variations**

Among all subsets of Euclidean space with a fixed volume, balls have the smallest perimeter. Furthermore, any set with nearly minimal perimeter is geometrically close, in a quantitative sense, to a ball. This latter statement reflects the quantitative stability of balls with respect to the perimeter functional. We will discuss recent advances in quantitative stability and applications in various contexts. The talk includes joint work with several collaborators and will be accessible to a broad research audience.

**Date**: November 18, 2022, 3:30 pm

**Place: **Pavillon André Aisenstadt, Room 5340, 2920, chemin de la tour, Montréal

Friday, November 11, 2022

**An equation of Diophantus**

This is the story of an old equation of Diophantus, which will take us on an excursion along a branch of number theory that stretches over a large part of the subject's history. It will be our motivation for a friendly introduction to some modern developments on torsion and ranks of elliptic curves.

**Date**: November 11, 2022, 3:30 pm

**Place: **Pavillon André Aisenstadt, Room 5340, 2920, chemin de la tour, Montréal

Friday, November 4, 2022

**Complexity of Submanifolds and Colding-Minicozzi Entropy**

Given a submanifold of Euclidean space, Colding and Minicozzi defined its entropy to be the supremum of the Gaussian weighted surface areas of all of its translations and dilations. While initially introduced to study singularities of mean curvature flow, it has proven to be an interesting geometric measure of complexity. In this talk I will survey some of the recent progress made on studying the Colding-Minicozzi entropy of hypersurfaces. In particular, I will discuss a series of work by Lu Wang and myself showing closed hypersurfaces with small entropy are simple in various senses.

**Date**: November 4, 2022, 3:30 pm

**Place: **Pavillon André Aisenstadt, Room 5340, 2920, chemin de la tour, Montréal

Friday, October 28, 2022

**Computer Model Emulation using Deep Gaussian Processes**

Computer models are often used to explore physical systems. Increasingly, there are cases where the model is fast, the code is not readily accessible to scientists, but a large suite of model evaluations is available. In these cases, an “emulator” is used to stand in for the computer model. This work was motivated by a simulator for the chirp mass of binary black hole mergers where no output is observed for large portions of the input space and more than 10^{6} simulator evaluations are available. This poses two problems: (i) the need to address the discontinuity when observing no chirp mass; and (ii) performing statistical inference with a large number of simulator evaluations. The traditional approach for emulation is to use a stationary Gaussian process (GP) because it provides a foundation for uncertainty quantification for deterministic systems. We explore the impact of the choices when setting up the deep GP on posterior inference, apply the proposed approach to the real application and propose a sequential design approach for identifying new simulations.

**Date**: October 28, 2022, 3:30 pm

**Place: **HEC, room Hélène-Desmarais, chemin de la Côte-Sainte-Catherine

Friday, October 14, 2022

I will describe two new isoperimetric inequalities for k-dimensional submanifolds of R^n or a Banach space. As a consequence of one we obtain a new systolic inequality that was conjectured by Larry Guth. As a consequence of another, we obtain an asymptotic formula for volumes of minimal submanifolds that was conjectured by Mikhail Gromov. The talk is based on joint works with Boris Lishak, Alexander Nabutovsky and Regina Rotman; Fernando Marques and Andre Neves; Larry Guth.

**Date**: October 14, 2022, 3:30 pm

**Place: **Pavillon André Aisenstadt, Room 1355, 2920, chemin de la tour, Montréal

**Register** for zoom link.

Friday, October 7, 2022

**Random plane geometry - a gentle introduction**

Consider Z^{2}, and assign a random length of 1 or 2 to each edge based on independent fair coin tosses. The resulting random geometry, first passage percloation, is conjectured to have a scaling limit.

Most random plane geometric models (including hidden geometries) should have the same scaling limit.

I will explain the basics of the limiting geometry, the "directed landscape", the central object in the class of models named after Kardar, Parisi and Zhang.

**Date**: October 7, 2022, 3:30 pm

**Place: **Pavillon André Aisenstadt, Room 6214, 2920, chemin de la tour, Montréal

**Register** for zoom link.

Friday, September 30, 2022

**(Prix SSC) Full likelihood inference for abundance from capture-recapture data: semiparametric efficiency and EM-algorithm**

Capture-recapture experiments are widely used to collect data needed to estimate the abundance of a closed population. To account for heterogeneity in the capture probabilities, Huggins (1989) and Alho (1990) proposed a semiparametric model in which the capture probabilities are modelled parametrically and the distribution of individual characteristics is left unspecified. A conditional likelihood method was then proposed to obtain point estimates and Wald-type confidence intervals for the abundance. Empirical studies show that the small-sample distribution of the maximum conditional likelihood estimator is strongly skewed to the right, which may produce Wald-type confidence intervals with lower limits that are less than the number of captured individuals or even negative.

In this talk, we present a full likelihood approach based on Huggins and Alho's model. We show that the null distribution of the empirical likelihood ratio for the abundance is asymptotically chi-square with one degree of freedom, and the maximum empirical likelihood estimator achieves semiparametric efficiency. We further propose an expectation–maximization algorithm to numerically calculate the proposed point estimate and empirical likelihood ratio function. Simulation studies show that the empirical-likelihood-based method is superior to the conditional-likelihood-based method: its confidence interval has much better coverage, and the maximum empirical likelihood estimator has a smaller mean square error.

**Date**: September 30, 2022, 3:30 pm

**Register** for zoom link.

Friday, September 23, 2022

**A story about pointwise ergodic theorems**

Pointwise ergodic theorems provide a bridge between the global behaviour of the dynamical system and the local combinatorial statistics of the system at a point. Such theorem have been proven in different contexts, but typically for actions of semigroups on a probability space. Dating back to Birkhoff (1931), the first known pointwise ergodic theorem states that for a measure-preserving ergodic transformation T on a probability space, the mean of a function (its global average) can be approximated by taking local averages of the function at a point x over finite sets in the forward-orbit of x, namely {x, Tx, ..., T^n x}. Almost a century later, we revisit Birkhoff's theorem and turn it backwards, showing that the averages along trees of possible pasts also approximate the global average. This backward theorem for a single transformation surprisingly has applications to actions of free groups, which we will also discuss. This is joint work with Jenna Zomback.

**Date**: September 23, 2022, 3:30 pm

**Place: **Pavillon André Aisenstadt, Room 5340, 2920, chemin de la tour, Montréal

**Register** for zoom link.

Friday, September 16, 2022

**Modularity of Galois representations, from Ramanujan to Serre's conjecture and beyond**

Ramanujan made a series of influential conjectures in his 1916 paper "On some arithmetical functions" on what is now called the Ramanujan τ\tauτ function. A congruence Ramanujan observed for τ(n)\tau(n)τ(n) modulo 691 in the paper led to Serre and Swinnerton-Dyer developing a geometric theory of mod ppp modular forms. It was in the context of the theory of mod ppp modular forms that Serre made his modularity conjecture, which was initially formulated in a letter of Serre to Tate in 1973.

I will describe the path from Ramanujan's work in 1916, to the formulation of a first version of Serre's conjecture in 1973, to its resolution in 2009 by Jean-Pierre Wintenberger and myself. I will also try to indicate why this subject is very much alive and, in spite of all the progress, still in its infancy.

**Date**: September 16, 2022, 3:30 pm

**Place: **Pavillon André Aisenstadt, Room 5340, 2920, chemin de la tour, Montréal

**Register** for zoom link.

Friday, September 2, 2022

**Algèbres amassées et théorie des noeuds**

Les algèbres amassées sont des algèbres de polynômes de Laurent dont les générateurs s’obtiennent par un processus récursif appelé mutation. On commence avec une graine, paire formée d’un ensemble de *n* variables, appelé amas, et d’un graphe orienté à *n* points. La mutation d’une graine remplace une variable à la fois et modifie le graphe, donnant ainsi une nouvelle graine. L’algèbre amassée est engendrée par toutes les variables obtenues par mutations successives, qu’on appelle variables amassées.

Les algèbres amassées ont été définies il y a 20 ans, et depuis, des liens entre elles et divers champs de recherche ont été découverts. Récemment, avec mon collaborateur, nous avons établi un lien entre la théorie des noeuds et les algèbres amassées. Cet exposé introduira d'abord les algèbres amassées puis présentera la connection entre ces dernières et le polynôme d'Alexander d'un noeud.

**Date**: September 2, 2022, 3:30 pm

**Place: **Pavillon André Ainsenstadt, room 6214/6254, Université de Montréal

Monday, July 4, 2022

The aim of this ISM discovery school is to explore different topics in representation theory through the lens of mutations.

UQAM, Montréal

July 4-8, 2022

Monday, June 6, 2022

There are relatively new developments connecting the geometry of Hessenberg varieties and symmetric functions and their associated combinatorics, which have shown that the (equivariant) geometry and topology of Hessenberg varieties are intimately connected with a deep unsolved problem in the theory of symmetric (and quasisymmetric) functions called the Stanley-Stembridge conjecture.

The subject of Hessenberg varieties lies in the fruitful intersection of algebraic geometry, combinatorics, and geometric representation theory. A fundamental contribution in this area, over a decade ago, was Julianna Tymoczko’s construction of an action of the symmetric group on the cohomology rings of regular semisimple Hessenberg varieties. Tymoczko’s action provided the first link between Hessenberg varieties and symmetric functions because representations of symmetric groups give rise to symmetric functions via the Frobenius characteristic map.

The second link was developed via the notion of the chromatic symmetric function of a graph, introduced by Richard Stanley in 1995 as a generalization of the classic chromatic polynomial of a graph. The Stanley-Stembridge conjecture concerns the structure of the chromatic symmetric functions of a special family of graphs; it states that the chromatic symmetric functions of these graphs are non-negative linear combinations of elementary symmetric functions. This conjecture is still open.

A close relationship between chromatic symmetric functions and Hessenberg varieties was discovered by John Shareshian and Michelle Wachs, who associated a graph with each regular semisimple Hessenberg variety and formulated a conjecture relating the chromatic symmetric function of the graph with the symmetric function associated with the Hessenberg variety via Tymoczko’s action. Their conjecture has since been proved, and there is reason to hope that further progress on the Stanley-Stembridge conjecture can be made by better understanding the relation between these two areas.

The Summer School is aimed at graduate students specializing in geometry or combinatorics, and the goal is to introduce both the theories of Hessenberg varieties and of symmetric functions in such a way that a student can have access to the exciting developments linking these areas.

Friday, May 27, 2022

The 24th edition of the Colloque Panquébécois de l’Institut des Sciences Mathématiques (ISM) will be held **in person at Université Laval** this **May 27-29, 2022**. The goal of this annual conference is to bring together graduate students in mathematics from all of Quebec’s universities.

Participants are invited to give a 20 minute talk on a mathematical subject of their choice. In addition, four plenary talks will be given by professors. The event will also feature a talk by the recipient of the Carl Herz prize, awarded by the ISM, and many social activities to give participants the opportunity to meet and network with other students in mathematics.

Friday, May 20, 2022

**Mathematical analysis of dilute gases: derivation of the Boltzmann equation, fluctuations and large deviations**

he evolution of a gas can be described by different models depending on the observation scale. A natural question, raised by Hilbert in his sixth problem, is whether these models provide consistent predictions. In particular, for rarefied gases, it is expected that continuum laws of kinetic theory can be obtained directly from molecular dynamics governed by the fundamental principles of mechanics. In the case of hard sphere gases, Lanford showed in 1975 that the Boltzmann equation emerges as the law of large numbers in the low density limit, at least for very short times. The goal of this talk is to explain the heuristics of his proof and present recent progress in the understanding of this limiting process.

**Date**: May 20, 2022, 3:30 pm

Friday, May 6, 2022

**Generic measure preserving transformations and descriptive set theory**

The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation has emerged. This picture included substantial evidence that pointed to these groups being all topologically isomorphic to a single group, namely, *L*^{0}---the non-locally compact, topological group of all Lebesgue measurable functions from [0,1] to the circle. In fact, Glasner and Weiss asked if this was the case.

We will describe the background touched on above, including the connections with Descriptive Set Theory. Further, we will indicate a proof of the following theorem that answers the Glasner--Weiss question in the negative: for a generic measure preserving transformation *T*, the closed group generated by *T* is **not** topologically isomorphic to *L*^{0}.

**Date**: May 6, 2022, 3:30 pm

Friday, April 29, 2022

**COVID-19 transmission models in the real world: models, data, and policy**

Simple mathematical models of COVID-19 transmission gained prominence in the early days of the pandemic. These models provided researchers and policymakers with qualitative insight into the dynamics of transmission and quantitative predictions of disease incidence. More sophisticated models incorporated new information about the natural history of COVID-19 disease and the interaction of infected individuals with the healthcare system, to predict diagnosed cases, hospitalization, ventilator usage, and death. Models also provided intuition for discussions about outbreaks, vaccination, and the effects of non-pharmaceutical interventions like social distancing guidelines and stay-at-home orders. But as the pandemic progressed, complex real-world interventions took effect, people everywhere changed their behavior, and the usefulness of simple mathematical models of COVID-19 transmission diminished. This challenge forced researchers to think more broadly about empirical data sources that could help predictive models regain their utility for guiding public policy. In this presentation, I will describe my view of the successes and failures of population-level transmission models in the context of the COVID-19 pandemic. I will outline the evolution of a project to predict COVID-19 incidence in the state of Connecticut, from development of a transmission model to engagement with public health policymakers and initiation of a new data collection effort. I argue that a new data source – passive measurement of close interpersonal contact via mobile device location data – is a promising way to overcome many of the shortcomings of traditional transmission models. I conclude with a summary of the impact this work has had on the COVID-19 response in Connecticut and beyond.

**Date**: April 29, 2022, 3:30 pm

Friday, April 22, 2022

**Cactus groups and monodromy**

The cactus group is a cousin of the braid group and shares many of its beautiful properties. It is the fundamental group of the moduli space of points on RP^{1}. It also acts on many collections of combinatorial objects. I will explain how we use the cactus group to understand monodromy of eigenvectors for Gaudin algebras.

This conference will be held in hybrid mode (on site and by Zoom).

**Date**: April 22, 2022, 3:30 pm

**Place: **Pavillon André Ainsenstadt, room 6214/6254, Université de Montréal

**For the Zoom link: Register**

Friday, April 15, 2022

**Some aspects of mean games**

Mean Field Game is the study of the dynamical behavior of a large number of agents in interaction. For instance, it can model be the dynamics of a crowd, or the production of a renewable resource by a large amount of producers. The analysis of these models, first introduced in the economic literature under the terminology of “heterogenous agent models, has known a spectacular development with the pioneering woks of Lasry and Lions and of Caines, Huang and Malhamé. The aim of the talk will be to illustrate the theory through a few models and present some of the main results and open questions.

**Date**: April 15, 2022, 3:00 pm

Friday, April 8, 2022

**Hidden Variable Model for Universal Quantum Computation with Magic States on Qubits**

We show that every quantum computation can be described by a probabilistic update of a probability distribution on a finite phase space. Negativity in a quasiprobability function is not required in states or operations, which is a very unusual feature. Nonetheless, our result is consistent with Gleason’s Theorem and the Pusey-Barrett- Rudolph theorem.

The reason I have chosen this subject for my talk is two-fold: (i) It gives the audience a glimpse of the quest to understand the quantum mechanical cause for speed-up in quantum computation, which is one of the central questions on the theory side of the field, and (ii) Maybe there can be feedback from the audience. The structures underlying the above probabilistic model are the so-called Lambda-polytopes, which are highly symmetric objects. At present we only know very few general facts about them. Help with analysing them would be appreciated!

Joint work with Michael Zurel and Cihan Okay,

Journal reference: Phys. Rev. Lett. 125, 260404 (2020)

**Date**: April 8, 2022, 3:30 pm

Friday, April 1, 2022

**Gentle algebras, surfaces and a glimpse of homological mirror symmetry**

Derived categories are in general not easy to parse. However, in certain cases, combinatorial models give a good picture of these categories. One such case are the bounded derived categories of gentle algebras which can be represented in terms of curves and crossings of curves on surfaces. In this talk, we will give the construction of these surface models and briefly explain how they are connected to the homological mirror symmetry programme. We will show how a combination of surface combinatorics and representation theory can give new insights into the associated categories.

**Date**: April 1, 2022, 2:00 pm

Friday, March 25, 2022

**Making mathematics computer-checkable**

In the last thirty years, computer proof verification became a mature technology, with successes including the checking of the Four-Colour Theorem, the Odd Order Theorem, and Hales' proof of the Kepler Conjecture. Recent advances such as the "Liquid Tensor Experiment" verifying a recent theorem of Scholze have provided further momentum, as likewise have promising experiments integrating this technology with machine learning.

I will briefly describe some of these developments. I will then try to describe, more generally, what it *feels* like to carry out research-level computer verifications of mathematics proofs: the level of expression one has access to, the ways one finds oneself interrogating and reorganizing a paper proof, the kinds of arguments which are more tedious (or less tedious!) than on paper.

**Date**: March 25, 2022, 3:30 pm

Friday, March 18, 2022

**The importance of large deviations in non-equilibrium systems**

Statistical Physics allowed to unify, at the end of the 19th century, Newton's mechanics and thermodynamics. It gave a way to predict the amplitude of fluctuations around the physical laws which were known at that time. Einstein, in his very first works, showed that the measurement of these fluctuations allowed to estimate the size of atoms. His reasoning, which was at the origin of the linear response theory, applied to the black body gave one of the first evidences of the duality wave-particle in Quantum Mechanics. Statistical Physics gives also a framework to predict large deviations for systems at equilibrium. In the last two decades, major efforts were devoted to extend our understanding of the statistical laws of fluctuations and large deviations to non-equilibrium systems. This talk will try to present some of the recent progresses.

**Date**: March 18, 2022, 3:30 pm

Monday, March 14, 2022

Happy International Math Day 2022 with the theme "Math Unites"! You can watch the 48-hour online coverage (March 14 around the world) and visit the best photos of the photo challenge.

Here are two events organized by the CRM.

1. Virtual Public Lecture by Francis Su (Harvey-Mudd College):

*Mathematics for Human Flourishing*

In particular, Francis Su has developed a friendship with a prisoner, Christopher, in a maximum security prison in the United States and this prisoner is waking up to mathematics.

2. This conference is preceded by the launch of a UNESCO tool kit that the CRM has developed entitled "Mathematics for Action: Supporting Science Based Decision" at 18:30 EDT. The kit can be viewed at: kit

The Canadian launch will be held on Monday, March 14, 2022, 18:30-19:15 EDT in hybrid mode. See program and registration

Friday, March 11, 2022

**Algebra, geometry and combinatorics of link homology**

Khovanov and Rozansky defined in 2005 a triply graded link homology theory which generalizes HOMFLY-PT polynomial. In this talk, I will outline some known results and structures in Khovanov-Rozansky homology, describe its connection to q,t-Catalan combinatorics and present several geometric models for some classes of links.

**Date**: March 11, 2022, 3:30 pm

Friday, February 18, 2022

**Structure learning for Extremal graphical models**

Extremal graphical models are sparse statistical models for multivariate extreme events. The underlying graph encodes conditional independencies and enables a visual interpretation of the complex extremal dependence structure. For the important case of tree models, we provide a data-driven methodology for learning the graphical structure. We show that sample versions of the extremal correlation and a new summary statistic, which we call the extremal variogram, can be used as weights for a minimum spanning tree to consistently recover the true underlying tree. Remarkably, this implies that extremal tree models can be learned in a completely non-parametric fashion by using simple summary statistics and without the need to assume discrete distributions, existence of densities, or parametric models for marginal or bivariate distributions. Extensions to more general graphs are also discussed.

**Date: F**ebruary 18, 2022, 3:30 pm

Friday, February 11, 2022

**Sticky particle dynamics**

I will discuss the time evolution of a collection of particles that interact primarily through perfectly inelastic collisions. I will explain why this problem is tractable if the particles are constrained to lie on a line versus if they are allowed to move freely in space. In particular, I'll also describe an equation at the heart of this difficulty which some researchers believe has been solved and others do not. This topic has motivations in astronomy and connections with optimal mass transportation which I will touch upon if time permits.

**Date: F**ebruary 11, 2022, 3:30 pm

Friday, February 4, 2022

**Euler Systems and the Birch--Swinnerton-Dyer conjecture**

L-functions are one of the central objects of study in number theory. There are many beautiful theorems and many more open conjectures linking their values to arithmetic problems. The most famous example is the conjecture of Birch and Swinnerton-Dyer, which is one of the Clay Millenium Prize Problems. I will discuss this conjecture and some related open problems, and I will describe some recent progress on these conjectures, using tools called "Euler systems".

**Date: F**ebruary 4, 2022, 12:00 pm

Friday, January 28, 2022

**Risk assessment, heavy tails, and asymmetric least squares techniques**

Statistical risk assessment, in particular in finance and insurance, requires estimating simple indicators to summarize the risk incurred in a given situation. Of most interest is to infer extreme levels of risk so as to be able to manage high-impact rare events such as extreme climate episodes or stock market crashes. A standard procedure in this context, whether in the academic, industrial or regulatory circles, is to estimate a well-chosen single quantile (or Value-at-Risk). One drawback of quantiles is that they only take into account the frequency of an extreme event, and in particular do not give an idea of what the typical magnitude of such an event would be. Another issue is that they do not induce a coherent risk measure, which is a serious concern in actuarial and financial applications. In this talk, after giving a leisurely tour of extreme quantile estimation, I will explain how, starting from the formulation of a quantile as the solution of an optimization problem, one may come up with two alternative families of risk measures, called expectiles and extremiles, in order to address these two drawbacks. I will give a broad overview of their properties, as well as of their estimation at extreme levels in heavy-tailed models, and explain why they constitute sensible alternatives for risk assessment using real data applications. This is based on joint work with Abdelaati Daouia, Irène Gijbels, Stéphane Girard, Simone Padoan and Antoine Usseglio-Carleve.

**Date: **January 28, 2022, 3:30 pm

Friday, January 21, 2022

**The commuting variety and generic pipe dreams**

Nobody knows whether the scheme "pairs of commuting nxn matrices" is reduced. I'll show how this scheme relates to matrix Schubert varieties, and give a formula for its equivariant cohomology class (and that of many other varieties) using "generic pipe dreams" that I'll introduce. These interpolate between ordinary and bumpless pipe dreams. With those, I'll rederive both formulae (ordinary and bumpless) for double Schubert polynomials. This work is joint with Paul Zinn-Justin.

**Date: **January 21, 2022, 2:00 pm

Friday, January 14, 2022

**Looking at hydrodynamics through a contact mirror: From Euler to Turing and beyond**

What physical systems can be non-computational? (Roger Penrose, 1989). Is hydrodynamics capable of calculations? (Cris Moore, 1991). Can a mechanical system (including the trajectory of a fluid) simulate a universal Turing machine? (Terence Tao, 2017).

The movement of an incompressible fluid without viscosity is governed by Euler equations. Its viscid analogue is given by the Navier-Stokes equations whose regularity is one of the open problems in the list of problems for the Millenium bythe Clay Foundation. The trajectories of a fluid are complex. Can we measure its levels of complexity (computational, logical and dynamical)?

In this talk, we will address these questions. In particular, we will show how to construct a 3-dimensional Euler flow which is Turing complete. Undecidability of fluid paths is then a consequence of the classical undecidability of the halting problem proved by Alan Turing back in 1936. This is another manifestation of complexity in hydrodynamics which is very different from the theory of chaos.

Our solution of Euler equations corresponds to a stationary solution or Beltrami field. To address this problem, we will use a mirror [5] reflecting Beltrami fields as Reeb vector fields of a contact structure. Thus, our solutions import techniques from geometry to solve a problem in fluid dynamics. But how general are Euler flows? Can we represent any dynamics as an Euler flow? We will address this universality problem using the Beltrami/Reeb mirror again and Gromov's h-principle. We will also consider the non-stationary case. These universality features illustrate the complexity of Euler flows. However, this construction is not "physical" in the sense that the associated metric is not the euclidean metric. We will announce an euclidean construction and its implications to complexity and undecidability.

These constructions [1,2,3,4] are motivated by Tao's approach to the problem of Navier-Stokes [7,8,9] which we will also explain.

[1] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. Universality of Euler flows and flexibility of Reeb embeddings. https://arxiv.org/abs/1911.01963.

[2] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. Constructing Turing complete Euler flows in dimension 3. Proc. Natl. Acad. Sci. 118 (2021) e2026818118.

[3] R. Cardona, E. Miranda, D. Peralta-Salas. Turing universality of the incompressible Euler equations and a conjecture of Moore. Int. Math. Res. Notices, , 2021;, rnab233, https://doi.org/10.109/imrn/rnab233

[4] R. Cardona, E. Miranda, D. Peralta-Salas. Computability and Beltrami fields in Euclidean space. https://arxiv.org/abs/2111.03559

[5] J. Etnyre, R. Ghrist. Contact topology and hydrodynamics I. Beltrami fields and the Seifert conjecture. Nonlinearity 13 (2000) 441–458.

[6] C. Moore. Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity 4 (1991) 199–230.

[7] T. Tao. On the universality of potential well dynamics. Dyn. PDE 14 (2017) 219–238.

[8] T. Tao. On the universality of the incompressible Euler equation on compact manifolds. Discrete Cont. Dyn. Sys. A 38 (2018) 1553–1565.

[9] T. Tao. Searching for singularities in the Navier-Stokes equations. Nature Rev. Phys. 1 (2019) 418–419.

**Date: **January 14, 2022, 11:00 am

Friday, December 17, 2021

**Nonparametric causal mediation in a time-to-event setting**

A causal mediation model with multiple time-to-event mediators is exemplified by the natural course of human disease marked by sequential milestones with a time-to-event nature. For example, from hepatitis B infection to death, patients may experience intermediate events such as liver cirrhosis and liver cancer. The sequential events of hepatitis, cirrhosis, cancer, and death are susceptible to right censoring; moreover, the latter events may preclude the former events. Casting the natural course of human diseases in the framework of causal mediation modeling, we establish a model with intermediate and terminal events as the mediators and outcomes, respectively. We define the interventional analog of path-specific effects (iPSEs) as the effect of an exposure on a terminal event mediated (or not mediated) by any combination of intermediate events without parametric models. The expression of a counting process-based counterfactual hazard is derived under the sequential ignorability assumption. We employ composite nonparametric likelihood estimation to obtain maximum likelihood estimators for the counterfactual hazard and iPSEs. Our proposed estimators achieve asymptotic unbiasedness, uniform consistency, and weak convergence. Applying the proposed method, we show that hepatitis B induced mortality is mostly mediated through liver cancer and/or cirrhosis whereas hepatitis C induced mortality may be through extrahepatic diseases.

**Date: **December 17, 2021, 10:00 am

Friday, December 10, 2021

**Stark's Conjectures and Hilbert's 12th Problem**

In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory and the special values of L-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field; this question lies at the core of Hilbert's 12th Problem. Meanwhile, there is an abundance of conjectures on the values of L-functions at certain special points. Of these, Stark's Conjecture has relevance toward explicit class field theory. I will describe two recent joint results with Mahesh Kakde on these topics. The first is a proof of the Brumer-Stark conjecture away from p=2. This conjecture states the existence of certain canonical elements in abelian extensions of totally real fields. The second is a proof of an exact formula for Brumer-Stark units that has been developed over the last 15 years. We show that these units together with other easily written explicit elements generate the maximal abelian extension of a totally real field, thereby giving a p-adic solution to the question of explicit class field theory for these fields.

**Date: **December 10, 2021, 2:00 pm

Friday, December 3, 2021

**K3 surfaces: geometry and dynamics**

K3 surfaces are a class of compact complex manifolds that enjoys many special properties and play an important role in several areas of mathematics. In this colloquium I will discuss a new interplay between complex geometry and analysis on K3 surfaces equipped with their Calabi-Yau metrics, and dynamics of holomorphic diffeomorphisms of these surfaces, that Simion Filip and I have been investigating recently.

**Date: **December 3, 2021, 3:30 pm

Friday, November 26, 2021

**Adventures with Partial Identifications in Studies of Marked Individuals**

Monitoring marked individuals is a common strategy in studies of wild animals (referred to as mark-recapture or capture-recapture experiments) and hard to track human populations (referred to as multi-list methods or multiple-systems estimation). A standard assumption of these techniques is that individuals can be identified uniquely and without error, but this can be violated in many ways. In some cases, it may not be possible to identify individuals uniquely because of the study design or the choice of marks. Other times, errors may occur so that individuals are incorrectly identified. I will discuss work with my collaborators over the past 10 years developing methods to account for problems that arise when are only individuals are only partially identified. I will present theoretical aspects of this research, including an introduction to the latent multinomial model and algebraic statistics, and also describe applications to studies of species ranging from the golden mantella (an endangered frog endemic to Madagascar measuring only 20 mm) to the whale shark (the largest known species of sh measuring up to 19 m).

**Date: **November 26, 2021, 3:30 pm

Friday, November 19, 2021

**Exploring string vacua through geometric transitions**

A fundamental problem in string theory is the multitude of distinct geometries which give rise to consistent solutions of the vacuum equations of motion. One possible resolution of this "vacuum degeneracy" problem is the "fantasy" that the moduli space of string vacua is connected through the process of "geometric transitions". I will discuss some geometric problems associated to this fantasy and their applications.

**Date: **November 19, 2021, 3:30 pm

Friday, November 12, 2021

**Estimating the mean of a random vector**

One of the most basic problems in statistics is the estimation of the mean of a random vector, based on independent observations. This problem has received renewed attention in the last few years, both from statistical and computational points of view. In this talk, we review some recent results on the statistical performance of mean estimators that allow heavy tails and adversarial contamination in the data. In particular, we are interested in estimators that have a near-optimal error in all directions in which the variance of the one dimensional marginal of the random vector is not too small. The material of this talk is based on a series of joint papers with Shahar Mendelson.

**Date: **November 12, 2021, 3:30 pm

**Place** : HYBRIDE

**This conference will be held in hybrid mode** with a **limited number of 21 participants** on site. To reserve your place by **first come, first serve**, please use the link below.

**On-site **: CRM - **Pavillon André Aisenstadt: Salle/ Room 5340**

https://docs.google.com/spreadsheets/d/1tUFNfKuoj_aoevoK9K7jvZS2W5dSi97m-BVIJdoMIgU/edit?usp=sharing

**Vaccination passport and ID will be required**

Friday, November 5, 2021

**Les mathématiques ont une histoire et une géographie**

La présentation se divise en deux temps. Dans un premier temps, les principaux résultats de notre étude sur le « portrait mathématique » des étudiants du Québec effectué dans le cadre du projet En avant math! (projet conjoint CRM-CIRANO soutenu par le Ministère des finances) seront présentés. Ce rapport se fonde d’une part sur les résultats des tests internationaux TIMS et PISA pour les élèves québécois du primaire et du secondaire et d’autre part, sur la situation des mathématiques dans les universités québécoises dégagée des données du Bureau de coopération interuniversitaire (BCI) ( évolution des inscriptions étudiantes et portrait des étudiants tant en genre qu’en statut). Les données du BCI montrent que l'étudiant typique inscrit en maths dans les universités québécoises est citoyen canadien, blanc et masculin. Et le nombre total d'inscrits baisse à chaque année (sauf, peut-être au doctorat). Où sont les filles? Où sont les étudiants issus de l'immigration récente? Et pourtant aux tests PISA et TIMMS les élèves issus de l'immigration performent mieux au Canada que les élèves canadiens (c'est l'inverse pour la moyenne des pays de l'OCDE).

Puis, dans un deuxième temps, à la lumière des résultats du portrait des étudiants, nous discuterons des enjeux sociaux pour des mathématiques plus inclusives. Une recherche collaborative avec les communautés inuit du Nunavik viendra illustrer nos propos.

**Date: **November 5, 2021, 3:30 pm

Friday, October 29, 2021

**Opinionated practices for teaching reproducibility: motivation, guided instruction and pratice**

In the data science courses at the University of British Columbia, we define data science as the study, development and practice of reproducible and auditable processes to obtain insight from data. While reproducibility is core to our definition, most data science learners enter the field with other aspects of data science in mind, for example predictive modelling, which is often one of the most interesting topic to novices. This fact, along with the highly technical nature of the industry standard reproducibility tools currently employed in data science, present out-ofthe gate challenges in teaching reproducibility in the data science classroom. Put simply, students are not as intrinsically motivated to learn this topic, and it is not an easy one for them to learn. What can a data science educator do? Over several iterations of teaching courses focused on reproducible data science tools and workflows, we have found that providing extra motivation, guided instruction and lots of practice are key to effectively teaching this challenging, yet important subject. Here we present examples of how we deeply motivate, effectively guide and provide ample practice opportunities to data science students to effectively engage them in learning about this topic.

**Date: **October 29, 2021, 3:30 pm

Friday, October 15, 2021

**Entropy along the Mandelbrot set**

The notion of topological entropy, arising from information theory, is a fundamental tool to understand the complexity of a dynamical system. When the dynamical system varies in a family, the natural question arises of how the entropy changes with the parameter.

In the last decade, W. Thurston introduced these ideas in the context of complex dynamics by defining the "core entropy" of a quadratic polynomials as the entropy of a certain forward-invariant set of the Julia set (the Hubbard tree).

As we shall see, the core entropy is a purely topological/combinatorial quantity which nonetheless captures the richness of the fractal structure of the Mandelbrot set. In particular, we will relate the variation of such a function to the geometry of the Mandelbrot set. We will also prove that the core entropy on the space of polynomials of a given degree varies continuously, answering a question of Thurston.

Finally, we will provide a new interpretation of core entropy in terms of measured laminations and discuss its finer regularity properties such as its Holder exponent.

**Date: **October 15, 2021, 3:30 pm

Friday, September 24, 2021

**Deep down, everyone wants to be causal**

Most researchers in the social, behavioral, and health sciences are taught to be extremely cautious in making causal claims. However, causal inference is a necessary goal in research for addressing many of the most pressing questions around policy and practice. In the past decade, causal methodologists have increasingly been using and touting the benefits of more complicated machine learning algorithms to estimate causal effects. These methods can take some of the guesswork out of analyses, decrease the opportunity for “p-hacking,” and may be better suited for more fine-tuned tasks such as identifying varying treatment effects and generalizing results from one population to another. However, should these more advanced methods change our fundamental views about how difficult it is to infer causality? In this talk I will discuss some potential advantages and disadvantages of using machine learning for causal inference and emphasize ways that we can all be more transparent in our inferences and honest about their limitations.

**Date: **September 24, 2021, 3:00 pm

Monday, June 21, 2021

**June 21-25, 2021****Fully Online**

The aim of this school is to advance the participants knowledge and enthusiasm towards algebraic combinatorics. Through high-level presentations, the students will learn multiple combinatorial aspects linked to representation theory. Every day, a postdoctoral researcher will introduce a research topic tied to the introductory classes.

Schubert calculus, symmetric functions, cluster algebra, Tamari lattices, frieze combinatorics and cluster categories are not only ways to study representation theory, but have many links between them. On one hand, cluster algebras, introduced by Sergey Fomin and Andrei Zelevinsky, can be studied using the combinatorics of friezes, on the other hand, they can be studied algebraically using cluster categories. Moreover, they have a correspondence with double Bruhat cells. In the case of Flag varieties and grassmanians, the decomposition into Bruhat cells gives way to decomposition into Schubert cells. These can be obtained using Schubert calculus. Schubbert polynomials are a generalization of Schur functions, which are symmetric functions. Using sub-word complexes, Schubert varieties are tied to the study of Tamari Lattices. These lattices correspond to exchange graphs of some cluster algebra.

Finally, our goal is to promote the visibility and accomplishment of women in mathematics. *Even though the school is open to people of all genders, only women were invited to give lectures and talks.* It seems important to us to give the occasion to students to interact accomplished women in mathematics, since they are underrepresented among teachers in mathematics in universities.

Monday, June 21, 2021

**Narrowing the Gap: Addressing Mathematical Inequity in Indigenous Education**

on Monday, June 21, 2021, National Indigenous Peoples Day

at 4:00 p.m. - 5:00 p.m. (Eastern time)

**SPEAKER: Melania Alvarez (Pacific Institute for the Mathematical Sciences) **

This lecture will be delivered in English..

ABSTRACT:

In order to positively narrow the educational gap between the Indigenous communities and the rest of the population, there needs to be a continuous and long-term intervention for change. By leaving behind the philosophy of reduced expectations, mathematical scientists and educators in Western Canada have introduced a variety of interesting and challenging programs. Our first step has been to build partnerships with elders and schools run by Indigenous communities, as well as with urban public schools with a high concentration of at-risk students and Indigenous Students. With their input and support, a variety of outreach programs have been implemented, which will be described in this talk.

Also visit the page of the CRM Equity, Diversity and Inclusion Committee

Monday, May 31, 2021

**May 31 - June 3, 2021**

The school will consist of four days of courses aimed primarily at upper undergraduate and MSc students who are interested in pursuing further university education, and curious about modern topics in statistics that they are unlikely to have encountered in their training.

Saturday, May 22, 2021

The 23rd edition of the Colloque Panquébécois de l’Institut des Sciences Mathématiques (ISM) will be held **online** **May 22-23, 2021**. The goal of this annual conference is to bring together graduate students in mathematics from all of Quebec’s universities.

Participants are invited to give a 20 minute talk on a subject of their interest within mathematics. In addition, four plenary talks will be given by professors. The conference will also feature a talk by the recipient of the Carl Herz prize, awarded by the ISM.

Friday, April 30, 2021

**Knots, polynomials and signatures**

After a historical introduction to knot theory, the talk will be centered around two knot invariants, the Alexander polynomial and the signature. The aim is to introduce a finite abelian group that controls their relationship, and to illustrate this by several examples. Using Seifert matrices, the geometric questions are translated into arithmetic ones.

**Date: **April 30, 2021, 3:00 pm

Friday, April 23, 2021

**Date: **April 23, 2021, 3:00 pm

Friday, April 16, 2021

**Reflected Brownian motion in a wedge: from probability theory to Galois theory of difference equations**

We consider a reflected Brownian motion in a two-dimensional wedge. Under standard assumptions on the parameters of the model (opening of the wedge, angles of the reflections on the axes, drift), we study the algebraic and differential nature of the Laplace transform of its stationary distribution. We derive necessary and sufficient conditions for this Laplace transform to be rational, algebraic, differentially finite or more generally differentially algebraic. These conditions are explicit linear dependencies among the angles involved in the definition of the model.

To prove these results, we start from a functional equation that the Laplace transform satisfies, to which we apply tools from diverse horizons. To establish differential algebraicity, a key ingredient is Tutte's invariant approach, which originates in enumerative combinatorics. To establish differential transcendence, we turn the functional equation into a difference equation and apply Galoisian results on the nature of the solutions to such equations.

This is a joint work with M. Bousquet-Mélou, A. Elvey Price, S. Franceschi and C. Hardouin (https://arxiv.org/abs/2101.01562).

**Date: **April 16, 2021, 3:00 pm

Friday, April 9, 2021

**Insect Flight from Newton's law to Neurons**

Why do animals move the way they do? Bacteria, insects, birds, and fish share with us the necessity to move so as to live. Although each organism follows its own evolutionary course, it also obeys a set of common laws. At the very least, the movement of animals, like that of planets, is governed by Newton’s law: All things fall. On Earth, most things fall in air or water, and their motions are thus subject to the laws of hydrodynamics. Through trial and error, animals have found ways to interact with fluid so they can float, drift, swim, sail, glide, soar, and fly. This elementary struggle to escape the fate of falling shapes the development of motors, sensors, and mind. Perhaps we can deduce parts of their neural computations by understanding what animals must do so as not to fall.

We have been seeking mechanistic explanations of the complex movement of insect flight. Starting from the Navier-Stokes equations governing the unsteady aerodynamics of flapping flight, we worked to build a theoretical framework for computing flight and for studying the control of flight. I will discuss our recent computational and experimental studies of the balancing act of dragonflies and fruit flies: how a dragonfly recovers from falling upside-down and how a fly balances in air. In each case, the physics of flight informs us about the neural feedback circuitries underlying their fast reflexes.

**Date: **April 9, 2021, 3:00 pm

Friday, March 19, 2021

Après avoir rappelé divers éléments sur l'histoire de la construction des schémas de Boltzmann sur réseau, nous présentons notre approche "ABCD", fondée sur le fait que le schéma numérique est exact pour l'équation d'advection avec les vitesses du réseau. Cette analyse asymptotique permet d'écrire aux différents ordres les équations aux dérivées partielles conservatives équivalentes au schéma. Un réglage de paramètres permet dans les bons cas une approximation précise des équations des fluides compressibles.

**Date: **March 19, 2021, 3:00 pm

Friday, March 12, 2021

With its 2021 theme “Mathematics for a Better World”, UNESCO's International Mathematics Day highlights the impact of mathematical sciences to face challenges in areas such as artificial intelligence, prediction models, climate change, screening, equitable sharing as well as improving the quality of life in many unexpected ways thanks to the combined knowledge of the scientists who work there. The CRM has brought together outstanding speaker-organizers who each explore this year's theme in their own way.

Friday, March 12, 2021

Informative selection, in which the distribution of response variables given that they are sampled is different from their distribution in the population, is pervasive in complex surveys. Failing to take such informativeness into account can produce severe inferential errors, including biased and inconsistent estimation of population parameters. While several parametric procedures exist to test for informative selection, these methods are limited in scope and their parametric assumptions are difficult to assess. We consider two classes of nonparametric tests of informative selection. The first class is motivated by classic nonparametric two-sample tests. We compare weighted and unweighted empirical distribution functions and obtain tests for informative selection that are analogous to Kolmogorov-Smirnov and Cramer-von Mises. For the second class of tests, we adapt a kernel-based learning method that compares distributions based on their maximum mean discrepancy. The asymptotic distributions of the test statistics are established under the null hypothesis of noninformative selection. Simulation results show that our tests have power competitive with existing parametric tests in a correctly specified parametric setting, and better than those tests under model misspecification. A recreational angling application illustrates the methodology.

This is joint work with Teng Liu, Colorado State University.

**Date: **March 12, 2021, 3:30 pm

Monday, March 8, 2021

This event, which we will be held on International Women's Day, aims to raise awareness of the career challenges experienced by women mathematicians during the pandemic.

It will also serve as an occasion to bring together the CRM community to honour the achievements of women in mathematics.

Four outstanding mathematicians will highlight their recent work, as well as the unusual circumstances that either led to it or challenged it. These talks will be followed by an informal panel discussion held in the evening, with a focus on the perspective of junior women mathematicians.

Friday, March 5, 2021

Filmed in Canada, Iran, and the United States, the beautiful documentary *Secrets of the Surface: The Mathematical Vision of Maryam Mirzakhani* examines the life and mathematical work of Maryam Mirzakhani, an Iranian immigrant to the United States who became a superstar in her field. In 2014, she was both the first woman and the first Iranian to be honored by mathematics’ highest prize, the Fields Medal.

Mirzakhani’s contributions are explained in the film by leading mathematicians and illustrated by animated sequences. Her mathematical colleagues from around the world, as well as former teachers, classmates, and students in Iran today, convey the deep impact of her achievements. She is a true inspiration.

**Date/Time: **March 5, 2021, 7:30 PM

**Length:** 60 min

Friday, February 26, 2021

**Analytic solutions to algebraic equations, and a conjecture of Kobayashi**

A projective algebraic variety is defined as the zero locus of a finite family of homogeneous polynomials. Over the field of complex numbers, the geometry of such varieties is governed to a large extent by the sign, in a suitable sense, of the Ricci curvature form. When this sign is negative, the variety is expected to exhibit certain hyperbolicity properties in the sense of Kobayashi - as well as further very deep number-theoretic properties that are mostly conjectural, in the arithmetic situation. In particular, all entire holomorphic curves drawn on it should be contained in a proper algebraic subvariety: this is a famous conjecture of Green-Griffiths and Lang. Following recent ideas of D. Brotbek, we will try to explain here a rather elementary proof of a related conjecture of Kobayashi, stating that a general algebraic hypersurface of sufficiently high degree is hyperbolic, i.e. does not contain any entire holomorphic curve.

**Date: **February 26, 2021, 3:00 pm

Friday, February 19, 2021

**Local smoothing for the wave equation**

The local smoothing problem asks about how much solutions to the wave equation can focus. It was formulated by Chris Sogge in the early 90s. Hong Wang, Ruixiang Zhang, and I recently proved the conjecture in two dimensions. In the talk, we will build up some intuition about waves to motivate the conjecture, and then discuss some of the obstacles and some ideas from the proof.

**Date: **February 19, 2021, 3:30 pm

Friday, February 12, 2021

**Spatio-temporal methods for estimating subsurface ocean thermal response to tropical cyclones**

Tropical cyclones (TCs), driven by heat exchange between the air and sea, pose a substantial risk to many communities around the world. Accurate characterization of the subsurface ocean thermal response to TC passage is crucial for accurate TC intensity forecasts and for understanding the role TCs play in the global climate system, yet that characterization is complicated by the high-noise ocean environment, correlations inherent in spatio-temporal data, relative scarcity of in situ observations and the entanglement of the TC-induced signal with seasonal signals. We present a general methodological framework that addresses these difficulties, integrating existing techniques in seasonal mean field estimation, Gaussian process modeling, and nonparametric regression into a functional ANOVA model. Importantly, we improve upon past work by properly handling seasonality, providing rigorous uncertainty quantification, and treating time as a continuous variable, rather than producing estimates that are binned in time. This functional ANOVA model is estimated using in situ subsurface temperature profiles from the Argo fleet of autonomous floats through a multi-step procedure, which (1) characterizes the upper ocean seasonal shift during the TC season; (2) models the variability in the temperature observations; (3) fits a thin plate spline using the variability estimates to account for heteroskedasticity and correlation between the observations. This spline fit reveals the ocean thermal response to TC passage. Through this framework, we obtain new scientific insights into the interaction between TCs and the ocean on a global scale, including a three-dimensional characterization of the near-surface and subsurface cooling along the TC storm track and the mixing-induced subsurface warming on the track's right side. Joint work with Addison Hu, Ann Lee, Donata Giglio and Kimberly Wood.

**Date: **February 12, 2021, 3:30 pm

Friday, February 5, 2021

**Symmetry, Barcodes, and Hamiltonian dynamics**

In the early 60s Arnol'd has conjectured that Hamiltonian diffeomorphisms, the motions of classical mechanics, often possess more fixed points than required by classical topological considerations. In the late 80s and early 90s Floer has developed a powerful theory to approach this conjecture, considering fixed points as critical points of a certain functional. Recently, in joint work with L. Polterovich, we observed that Floer theory filtered by the values of this functional fits into the framework of persistence modules and their barcodes, originating in data sciences. I will review these developments and their applications, which arise from a natural time-symmetry of Hamiltonians. This includes new constraints on one-parameter subgroups of Hamiltonian diffeomorphisms, as well as my recent solution of the Hofer-Zehnder periodic points conjecture. The latter combines barcodes with equivariant cohomological operations in Floer theory recently introduced by Seidel to form a new method with further consequences.

**Date: **February 5, 2021, 3:00 pm

Friday, January 29, 2021

**Small Area Estimation in Low- and Middle-Income Countries**

The under-five mortality rate (U5MR) is a key barometer of the health of a nation. Unfortunately, many people living in low- and middle-income countries are not covered by civil registration systems. This makes estimation of the U5MR, particularly at the subnational level, difficult. In this talk, I will describe models that have been developed to produce the official United Nations (UN) subnational U5MR estimates in 22 countries. Estimation is based on household surveys, which use stratified, two-stage cluster sampling. I will describe a range of area- and unit-level models and describe the rationale for the modeling we carry out. Data sparsity in time and space is a key challenge, and smoothing models are vital. I will discuss the advantages and disadvantages of discrete and continuous spatial models, in the context of estimation at the scale at which health interventions are made. Other issues that will be touched upon include: design-based versus model-based inference; adjustments for HIV epidemics; the inclusion of so-called indirect (summary birth history) data; reproducibility through software availability; benchmarking; how to deal with incomplete geographical data; and working with the UN to produce estimates.

**Date: **January 29, 2021, 3:00 pm

Friday, January 22, 2021

**Mean curvature flow through neck-singularities**

A family of surfaces moves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow first arose as a model of evolving interfaces and has been extensively studied over the last 40 years. In this talk, I will give an introduction and overview for a general mathematical audience. To gain some intuition we will first consider the one-dimensional case of evolving curves. We will then discuss Huisken’s classical result that the flow of convex surfaces always converges to a round point. On the other hand, if the initial surface is not convex we will see that the flow typically encounters singularities. Getting a hold of these singularities is crucial for most striking applications in geometry, topology and physics. Specifically, singularities can be either of neck-type or conical-type. We will discuss examples from the 90s, which show, both experimentally and theoretically, that flow through conical singularities is utterly non-unique. In the last part of the talk, I will report on recent work with Kyeongsu Choi, Or Hershkovits and Brian White, where we proved that mean curvature flow through neck-singularities is unique. The key for this is a classification result for ancient asymptotically cylindrical flows that describes all possible blowup limits near a neck-singularity. In particular, this confirms the mean-convex neighborhood conjecture. Assuming Ilmanen’s multiplicity-one conjecture, we conclude that for embedded two-spheres mean curvature flow through singularities is well-posed.

**Date: **January 22, 2021, 3:00 pm

Saturday, January 9, 2021

The Seminars on Undergraduate Mathematics in Montreal (SUMM) is organized by undergraduate students from Montreal universities. The main objective is to create an environment facilitating exchange of ideas and interests as well as allowing students to network. For the twelfth edition, we aim to unite the undergraduate mathematic community in Montreal, Quebec, and their surroundings.

SUMM is for undergraduate students in mathematics or related domains. This year, the conference will be held on **January 9 and 10**, online due to the pandemic.

The weekend consist of two days of presentations given by undergraduate students and invited professors. The presentations will covers a broad range of subject from mathematical physics to the applications of artificial intelligence as well as the history and philosophy of mathematics.

During the SUMM, students can give a talk or simply attend presentations from their peers. It's an occasion to share the passion for mathematics in a stimulating environment, while networking with other passionate students over the weekend.

Friday, November 27, 2020

**Moduli of unstable objects in algebraic geometry**Moduli spaces arise naturally in classification problems in geometry. The study of the moduli spaces of nonsingular complex projective curves (or equivalently of compact Riemann surfaces) goes back to Riemann himself in the nineteenth century. The construction of the moduli spaces of stable curves of fixed genus is one of the classical applications of Mumford's geometric invariant theory (GIT), developed in the 1960s; many other moduli spaces of 'stable' objects can be constructed using GIT and in other ways. A projective curve is stable if it has only very mild singularities (nodes) and its automorphism group is finite; similarly in other contexts stable objects are usually better behaved than unstable ones.

The aim of this talk is to explain how recent methods from a version of GIT for non-reductive group actions can help us to classify singular curves in such a way that we can construct moduli spaces of unstable curves (of fixed type). More generally our aim is to use suitable 'stability conditions' to stratify other moduli stacks into locally closed strata with coarse moduli spaces. The talk is based on joint work with Gergely Berczi, Vicky Hoskins and Joshua Jackson.

**Date: **November 27, 2020, 3:00 pm

Friday, November 20, 2020

**Hodge Theory of p-adic varieties**

p-adic Hodge Theory is one of the most powerful tools in modern Arithmetic Geometry. In this talk, I will review p-adic Hodge Theory of algebraic varieties, present current developments in p-adic Hodge Theory of analytic varieties, and discuss some of its applications to problems in Number Theory.

**Date: **November 20, 2020, 3:00 pm

Friday, November 13, 2020

**Approximate Cross-Validation for Large Data and High Dimensions**

The error or variability of statistical and machine learning algorithms is often assessed by repeatedly re-fitting a model with different weighted versions of the observed data. The ubiquitous tools of cross-validation (CV) and the bootstrap are examples of this technique. These methods are powerful in large part due to their model agnosticism but can be slow to run on modern, large data sets due to the need to repeatedly re-fit the model. We use a linear approximation to the dependence of the fitting procedure on the weights, producing results that can be faster than repeated re-fitting by orders of magnitude. This linear approximation is sometimes known as the "infinitesimal jackknife" (IJ) in the statistics literature, where it has mostly been used as a theoretical tool to prove asymptotic results. We provide explicit finite-sample error bounds for the infinitesimal jackknife in terms of a small number of simple, verifiable assumptions. Without further modification, though, we note that the IJ deteriorates in accuracy in high dimensions and incurs a running time roughly cubic in dimension. We additionally show, then, how dimensionality reduction can be used to successfully run the IJ in high dimensions when data is sparse or low rank. Simulated and real-data experiments support our theory.

**Date: **November 13, 2020, 3:30 pm

Friday, October 16, 2020

**Hyperplane arrangements and modular symbols**In his fantastic book « Elliptic functions according to Eisenstein and Kronecker, » Weil writes: « As Eisenstein shows, his method for constructing elliptic functions applies beautifully to the simpler case of the trigonometric functions. Moreover, this case provides […] the simplest proofs for a series of results, originally discovered by Euler. » The results Weil alludes to are relations between product of trigonometric functions. I will first explain how these relations are quite surprisingly governed by relations between modular symbols (whose elementary theory I will sketch). I will then show how this story fits into a wider picture that relates the topological world of group homology of some linear groups to the algebraic world of trigonometric and elliptic functions. To conclude I will briefly describe a number theoretical application. This is based on a work-in-progress with Pierre Charollois, Luis Garcia and Akshay Venkatesh.

**Date: **October 16, 2020, 3:00 PM

Friday, October 9, 2020

**Hodge Theory and Moduli**

The theory of moduli is an important and active area in algebraic geometry. For varieties of general type the existence of a moduli space* ** *** **with a canonical completion

In this talk, we will discuss some aspects of this topic with emphasis on I-surfaces, which provide one of the first examples where the theory has been worked out in some detail. Particular notice will me made of how the extension data in the limiting mixed Hodge structures that arise from singular surfaces on the boundary of moduli may be used to guide the desingularization of that boundary.

**Date: **October 9, 2020, 3:00 PM

Friday, October 2, 2020

**Data Science, Classification, Clustering and Three-Way Data**

Data science is discussed along with some historical perspective. Selected problems in classification are considered, either via specific datasets or general problem types. In each case, the problem is introduced before one or more potential solutions are discussed and applied. The problems discussed include data with outliers, longitudinal data, and three-way data. The proposed approaches are generally mixture model-based.

Zoom: If you haven't already, to receive the zoom link for the series please register at: http://crm.umontreal.ca/quebec-mathematical-sciences-colloquium/index.html#csmq

Friday, September 11, 2020

**Machine Learning for Causual Inference**

**Date: **September 11, 2020, 4:00 pm

Friday, June 19, 2020

**Quantitative approaches to understanding the immune response to SARS-CoV-2 infection**

COVID-19 is typically characterized by a range of respiratory symptoms that, in severe cases, progress to acute respiratory distress syndrome (ARDS). These symptoms are also frequently accompanied by a range of inflammatory indications, particularly hyper-reactive and dysregulated inflammatory responses in the form of cytokine storms and severe immunopathology. Much remains to be uncovered about the mechanisms that lead to disparate outcomes in COVID-19. Here, quantitative approaches, especially mechanistic mathematical models, can be leveraged to improve our understanding of the immune response to SARS-CoV-2 infection. Building upon our prior work modelling the production of innate immune cell subsets and the viral dynamics of HIV and oncolytic viruses, we are developing a quantitative framework to interrogate open questions about the innate and adaptive immune reaction in COVID-19. In this talk, I will outline our recent work modelling SARS-CoV-2 viral dynamics and the ensuing immune response at both the tissue and systemic levels. A portion of this work is done as part of an international and multidisciplinary coalition working to establish a comprehensive tissue simulator (physicell.org/covid19 [1]), which I will also discuss in more detail.

**Date: **June 19, 2020, 4:00 pm

Friday, April 17, 2020

**Observable events and typical trajectories in finite and infinite dimensional dynamical systems**

The terms "observable events" and "typical trajectories" in the title should really be between quotation marks, because what is typical and/or observable is a matter of interpretation. For dynamical systems on finite dimensional spaces, one often equates observable events with positive Lebesgue measure sets, and invariant distributions that reflect the large-time behaviors of positive Lebesgue measure sets of initial conditions (such as Liouville measure for Hamiltonian systems) are considered to be especially important. I will begin by introducing these concepts for general dynamical systems -- including those with attractors -- describing a simple dynamical picture that one might hope to be true. This picture does not always hold, unfortunately, but a small amount of random noise will bring it about. In the second part of my talk I will consider infinite dimensional systems such as semi-flows arising from dissipative evolutionary PDEs. I will discuss the extent to which the ideas above can be generalized to infinite dimensions, and propose a notion of "typical solutions".

**Date / Time**: Friday, April 17, 2020 - 16:00 ** **

**Venue**: Zoom meeting link:

https://umontreal.zoom.us/j/170851981?pwd=b1ZxMWM0Z3Q0d3I5ZHJUS0FUZEY5QT09

Meeting ID: 170 851 981

Password: 942210

Sunday, March 15, 2020

Filmed in Canada, Iran, and the United States, Secrets of the Surface: The Mathematical Vision of Maryam Mirzakhani examines the life and mathematical work of Maryam Mirzakhani, an Iranian immigrant to the United States who became a superstar in her field. In 2014, she was both the first woman and the first Iranian to be honored by mathematics’ highest prize, the Fields Medal.

Mirzakhani’s contributions are explained in the film by leading mathematicians and illustrated by animated sequences. Her mathematical colleagues from around the world, as well as former teachers, classmates, and students in Iran today, convey the deep impact of her achievements. The path of her education, success on Iran’s Math Olympiad team, and her brilliant work, make Mirzakhani an ideal role model for girls looking toward careers in science and mathematics.

**When**

15/03/2020

17:00-19:00.

**Where**

J.A. De Sève Cinema

Room 125

J.W. McConnell Building

1400 De Maisonneuve W.

Sir George Williams Campus

This event is free.

Friday, March 13, 2020

**Optimal shapes arising from pair interactions**

In many physical and social situations, pair interactions determine how a large group of particles or individuals arranges itself in space under constraints on the overall mass, density, and geometry. Typical examples are capacitor problems (where the interaction is purely repulsive), and flocking (where the interaction tends to be attractive at large distances and repulsive as individuals get too close). Mathematically, this leads to non-local shape optimization problems, where a density interacts with itself by a pair potential. Under what conditions is there aggregation, and when do individuals disperse? Is the optimal shape always round? Can multiple flocks co-exist? I will discuss some toy models, symmetrization techniques, recent results, and open questions.

**Date / Time**: Friday, March 13, 2020 - 4:00 pm** **

**Venue**: CRM, Université de Montréal, André-Aisenstadt Building, room 1175

Friday, February 28, 2020

**Neyman-Pearson classification: parametrics and sample size requirement**

The Neyman-Pearson (NP) paradigm in binary classification seeks classifiers that achieve a minimal type II error while enforcing the prioritized type I error controlled under some user-specified level alpha. This paradigm serves naturally in applications such as severe disease diagnosis and spam detection, where people have clear priorities among the two error types. Recently, Tong, Feng and Li (2018) proposed a nonparametric umbrella algorithm that adapts all scoring-type classification methods (e.g., logistic regression, support vector machines, random forest) to respect the given type I error (i.e., conditional probability of classifying a class 0 observation as class 1 under the 0-1 coding) upper bound alpha with high probability, without specific distributional assumptions on the features and the responses. Universal the umbrella algorithm is, it demands an explicit minimum sample size requirement on class 0, which is often the more scarce class, such as in rare disease diagnosis applications. In this work, we employ the parametric linear discriminant analysis (LDA) model and propose a new parametric thresholding algorithm, which does not need the minimum sample size requirements on class 0 observations and thus is suitable for small sample applications such as rare disease diagnosis. Leveraging both the existing nonparametric and the newly proposed parametric thresholding rules, we propose four LDA-based NP classifiers, for both low- and high-dimensional settings. On the theoretical front, we prove NP oracle inequalities for one proposed classifier, where the rate for excess type II error benefits from the explicit parametric model assumption. Furthermore, as NP classifiers involve a sample splitting step of class 0 observations, we construct a new adaptive sample splitting scheme that can be applied universally to NP classifiers, and this adaptive strategy reduces the type II error of these classifiers. The proposed NP classifiers are implemented in the R package nproc.

**Date / Time**: Friday, February 28, 2020 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, February 21, 2020

**Arithmetic Theta Series**

I will recount a family history of theta series through several generations. Theta series for positive definite integral quadratic forms provide some of the most classical examples of elliptic modular forms and their Siegel modular variants. Analogous series were defined by Siegel and Maass for lattices with indefinite quadratic forms say with signature (p,q). These series are no longer holomorphic and depend on an additional variable in the Grassmannian of negative q-planes, i.e., the symmetric space for the orthogonal group O(p,q). Motivated by work of Hirzebruch and Zagier on the generating series for curves on Hilbert modular surfaces, Millson and I constructed a theory of theta series valued in the cohomology of certain locally symmetric spaces -- geometric theta series. More recently, a theory of arithmetic theta series has been emerging, theta series valued in the Chow groups or arithmetic Chow groups of the integral models of certain Shimura varieties.

**Date / Time**: Friday, February 21, 2020 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, February 7, 2020

**Complex multiplication - old and new**

The theory of complex multiplication is more than a century old; its origins date back to Klein, Hilbert, Kummer, Weber, Deuring and many others. It has been instrumental in the development of class field theory and algebraic number theory. Yet, more than a century later we find new theorems that are truly surprising. I will start with this historical perspective and try to position some of these new developments in the light of the André-Oort conjecture - a conjecture in the area of Shimura varieties that was recently resolved by Tsimerman, building on ideas of Edixhoven, Pila, Wilkie and Zannier. The resolution rests on the averaged Colmez conjecture, a conjecture that addresses the arithmetic complexity of abelian varieties with complex multiplication, which was proved by Andreatta-Howard-Madapusi Pera and the speaker, and, independently, by Yuan-Zhang.

**Date / Time**: Friday, February 7, 2020 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, January 31, 2020

**Longitudinal functional regression: tests of significance **

We consider longitudinal functional regression, where, for each subject, the response consists of multiple curves observed at different time visits. We discuss tests of significance in two general settings. First, when there are no additional covariates, we develop a hypothesis testing methodology for formally assessing that the mean function does not vary over time. Second, in the presence of other covariates, we propose a testing procedure to determine the significance of the covariate's time-varying effect formally. The methods account for the complex dependence structure of the response and are computationally efficient. Numerical studies confirm that the testing approaches have the correct size and are have a superior power relative to available competitors. We illustrate the methods on a real data application.

**Date / Time**: Friday, January 31, 2020 - 16:00 ** **

**Venue**: HEC Montréal, 3000, chemin de la Côte-Sainte-Catherine, room Béton Grilli

Friday, January 24, 2020

**Propagation des ondes et diffraction par un obstacle: résultats utilisant l’analyse microlocale **

**Date / Time**: Friday, January 24, 2020 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, January 17, 2020

**Learning in Games**

Selfish behavior can often lead to suboptimal outcome for all participants, a phenomenon illustrated by many classical examples in game theory. Over the last decade we developed good understanding on how to quantify the impact of strategic user behavior on the overall performance in many games (including traffic routing as well as online auctions). In this talk we will focus on games where players use a form of learning that helps them adapt to the environment, and consider two closely related questions: What are broad classes of learning behaviors that guarantee high social welfare in games, and are these results robust to situations when game or the population of players is dynamically changing.

**Date / Time**: Friday, January 17, 2020 - 2:00 pm** **

**Venue**: CRM, Université de Montréal, André-Aisenstadt Building, room 6214-6254

Friday, January 10, 2020

The Seminar on Undergraduate Mathematics in Montreal (SUMM), is organized by undergraduate students from Montreal universities. The main objective is to create an environment facilitating exchange of ideas and interests as well as allowing students to network. For the eleventh edition, we aim to unite the undergraduate mathematic community in Montreal, Quebec, and their surroundings.

SUMM is for undergraduate student in mathematics or related domains. This year, the conference will be held on January 10, 11, and 12 at UQAM.

The weekend will start with a wine and cheese on Friday evening and will be followed by two days of presentations given by undergraduate students and six invited professors. The presentations will covers a broad range of subject from mathematical physic to the applications of artificial intelligence as well as the history and philosophy of mathematics.

During the SUMM, students can give a presentation, participate in a poster contest or simply attend presentations from their peers. It's an occasion to share the passion for mathematics in a stimulating environment, while networking with other passionate students over the weekend whether it is between presentations or during breakfast, lunch, or dinner and enjoying, for example, a delicious chicken panini.

Hope to see you there!

Friday, November 29, 2019

**Shuffling and Group Representations**

Picture *n* cards, numbered 1,2,...,*n* face down, in order, in a row on the table. Each time, your left hand picks a random card, your right hand picks a random card and the two cards are transposed. It is clear that 'after a while' the cards get all mixed up. How long does this take? In joint work with Mehrdad Shahshahani we analyzed this problem using the character theory of the symmetric group. The methods work for general measures on general compact groups. They mix probability, analysis, combinatorics and group theory (we need real formulas for the representations). I will try to explain all of this(along with some motivation for studying such problems) 'in English'. The answer, when *n*=52, is 'about 400'.

**Date / Time**: Friday, November 29, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, November 22, 2019

**Formulation and solution of stochastic inverse problems for science and engineering models**

The stochastic inverse problem of determining probability structures on input parameters for a physics model corresponding to a given probability structure on the output of the model forms the core of scientific inference and engineering design. We describe a formulation and solution method for stochastic inverse problems that is based on functional analysis, differential geometry, and probability/measure theory. This approach yields a computationally tractable problem while avoiding alterations of the model like regularization and ad hoc assumptions about the probability structures. We present several examples, including a high-dimensional application to determination of parameter fields in storm surge models. We also describe work aimed at defining a notion of condition for stochastic inverse problems and tackling the related problem of designing sets of optimal observable quantities.

**Date / Time**: Friday, November 22, 2019 - 16:00 ** **

**Venue**: UQAM, Pavillon Président-Kennedy - 201, avenue du Président-Kennedy, room PK-5115

Friday, November 15, 2019

**The role of random models in compressive sensing and matrix completion**

Random models lead to a precise and comprehensive theory of compressive sensing and matrix completion. The number of random linear measurements needed to recover a sparse signal, or a low-rank matrix, or, more generally, a structured signal, are now well understood. Indeed, this boils down to a question in random matrix theory: How well conditioned is a random matrix restricted to a fixed subset of R^{n}? We discuss recent work addressing this question in the sub-Gaussian case. Nevertheless, a practitioner with a fixed data set will wonder: Can they apply theory based on randomness? Is there any hope to get the same guarantees? We discuss these questions in compressive sensing and matrix completion, which, surprisingly, seem to have divergent answers.

**Date / Time**: Friday, November 15, 2019 - 16:00 ** **

**Venue**: CRM, Université de Montréal, André-Aisenstadt Building, room 135

Friday, November 8, 2019

**Symmetries in topological quantum field theories**

In this talk I will describe how to characterize symmetries of topological field theories and give a complete classification of symmetries in Abelian Topological Field Theories, uncovering a plethora of quantum symmetries in these theories and an intriguing connection to number theory.

**Date / Time**: Friday, November 8, 2019 - 16:00 ** **

**Venue**: CRM, Université de Montréal, André-Aisenstadt Building, room 1355

Friday, November 1, 2019

**General Bayesian modeling**

The work is motivated by the inflexibility of Bayesian modeling; in that only parameters of probability models are required to be connected with data. The idea is to generalize this by allowing arbitrary unknowns to be connected with data via loss functions. An updating process is then detailed which can be viewed as arising in at least a couple of ways - one being purely axiomatically driven. The further exploration of replacing probability model based approaches to inference with loss functions is ongoing. Joint work with Chris Holmes, Pier Giovanni Bissiri and Simon Lyddon.

**Date / Time**: Friday, November 1, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, October 25, 2019

**Coincidences in homological densities**

For certain natural sequences of topological spaces, the kth homology group stabilizes once you go far enough out in the sequence of spaces. This phenomenon is called homological stability. Two classical examples of homological stability are the configuration space of n unordered distinct points in the plane, studied in the 60's by Arnold' and the space of (based) algebraic maps from CP^{1} to CP^{1} studied by Segal in the 70's. It turns out that the stable homology is the same in these two examples, and in this talk we explain that this is just the tip an iceberg--a subtle, but precise relationship between the values of stable of homology different sequences of spaces. To explain this relationship, which we discovered through an analogy to asymptotic counts in number theory, we introduce a new notion of homological density. This talk is on joint work with Benson Farb and Jesse Wolfson.

**Date / Time**: Friday, October 25, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, October 18, 2019

**o-minimal GAGA and applications to Hodge theory**

(Joint with B.Bakker and Y.Brunebarbe) One very fruitful way of studying complex algebraic varieties is by forgetting the underlying algebraic structure, and just thinking of them as complex analytic spaces. To this end, it is a natural and fruitful question to ask how much the complex analytic structure remembers. One very prominent result is Chows theorem, stating that any closed analytic subspace of projective space is in fact algebraic. One notable consequence of this result is that a compact complex analytic space admits at most 1 algebraic structure - a result which is false in the non-compact case. This was generalized and extended by Serre in his famous GAGA paper using the language of cohomology. We explain how we can extend Chow's theorem and in fact all of GAGA to the non-compact case by working with complex analytic structures that are "tame" in the precise sense defined by o-minimality. This leads to some very general "algebraization" theorems, and we give applications to Hodge theory.

**Date / Time**: Friday, October 18, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, October 11, 2019

**Scoring positive semidefinite cutting planes for quadratic optimization via trained neural networks**

Semidefinite programming relaxations complement polyhedral relaxations for quadratic optimization, but global optimization solvers built on polyhedral relaxations cannot fully exploit this advantage. We develop linear outer-approximations of semidefinite constraints that can be effectively integrated into global solvers for nonconvex quadratic optimization. The difference from previous work is that our proposed cuts are (i) sparser with respect to the number of nonzeros in the row and (ii) explicitly selected to improve the objective. A neural network estimator is key to our cut selection strategy: ranking each cut based on objective improvement involves solving a semidefinite optimization problem, but this is an expensive proposition at each Branch&Cut node. The neural network estimator, trained a priori of any instance to solve, takes the most time consuming computation offline by predicting the objective improvement for any cut.

**Date / Time**: Friday, October 11, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, October 4, 2019

**La queue, la tuile, le bris d’égalité et leur rôle dans les modèles de dépendance**

La modélisation de la dépendance entre variables aléatoires est omniprésente en statistique. S’agissant d’événements rares à fort impact, tels que des orages violents, des inondations ou des vagues de chaleur, la question revêt une grande importance pour la gestion des risques et pose des défis théoriques. Une approche hautement flexible et prometteuse s’appuie sur la théorie des valeurs extrêmes, la modélisation par copules et l’inférence fondée sur les rangs. Je présenterai trois avancées récentes dans ce domaine. Nous nous intéresserons d’abord à la prise en compte de la dépendance en régime moyen, lorsque les modèles asymptotiques de valeurs extrêmes ne conviennent pas. Nous verrons ensuite quoi faire lorsque le nombre de variables est grand et comment une structure de modèle hiérarchique peut être apprise à partir de matrices de corrélation de rangs de grande taille. Enfin, je ne résisterai pas à l’envie de vous initier à l’univers complexe de l’inférence basée sur les rangs pour les données discrètes ou mixtes.

**Date / Time**: Friday, October 4, 2019 - 16:00 ** **

**Venue**: CRM, Université de Montréal, Pavillon André-Aisenstadt, room 1355

Thursday, September 26, 2019

**From Monge optimal transports to optimal Skorokhod embeddings**

The optimal transportation problem, which originated in the work of Gaspard Monge in 1781, provides a fundamental and quantitave way to measure the distance between probability distributions. It has led to many successful applications in PDEs, Geometry, Statistics and Probability Theory. Recently, and motivated by problems in Financial Mathematics, variations on this problem were introduced by requiring the transport plans to abide by certain "fairness rules," such as following martingale paths. One then specifies a stochastic state process and a costing procedure, and minimize the expected cost over stopping times with a given state distribution. Recent work has uncovered deep connections between this type of constrained optimal transportation problems, the celebrated Skorokhod embeddings of probability distributions in Brownian motion, and Hamilton-Jacobi variational inequalities.

**Date / Time**: Thursday, September 26, 2019 - 16:00 ** **

**Venue**: CRM, Université de Montréal, Pav. Roger-Gaudry, 2900, boul. Édouard-Montpetit, room M-415

Friday, September 20, 2019

**Date / Time**: Friday, September 20, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, September 13, 2019

**Multiple zeta values in deformation quantization**

The subject of "deformation quantization" originated as a mathematical abstraction of the passage from classical to quantum mechanics: starting from a classical phase space (i.e. a Poisson manifold), we deform the ordinary multiplication of functions to produce a noncommutative ring, which serves as the algebra of quantum observables. Over the years, the theory has evolved from its physical origins to take on a mathematical life of its own, with rich connections to representation theory, topology, graph theory, number theory and more. I will give an introduction to the subject and explain how the quantization process is inextricably linked, via a formula of Kontsevich, to special values of the Riemann zeta function, and their generalizations known as multiple zeta values.

**Date / Time**: Friday, September 13, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, August 30, 2019

**Khovanov homology, 3-manifolds, and 4-manifolds**

Khovanov homology is an invariant of knots in R^{3}. A major open problem is to extend its definition to knots in other three-manifolds, and to understand its relation to surfaces in 4-manifolds. I will discuss some partial progress in these directions, from different perspectives (gauge theory, representation theory, sheaf theory). In the process I will also review some of the topological applications of Khovanov homology.

**Date / Time**: Friday, August 30, 2019 - 16:00

** Venue**: UQAM, Pavillon Sherbrooke, 201, rue Sherbrooke West, room SH-3620

Tuesday, August 27, 2019

This discovery school is based around recent developments in low-dimensional topology, centred on the work of Ciprian Manolescu who will be in residence as an Aisenstadt chair as part of the CRM’s 50th anniversary program in Low dimensional topology. The school will be focused on new developments in Heegaard-Floer homology and gauge theory and the developments growing out of Manolescu’s celebrated disproof of the triangulation conjecture.

Monday, June 10, 2019

The analysis of dynamical (or time-dependent) stochastic processes and their asymptotic behaviour is fundamental in both discrete and continuous probability. In some cases, the asymptotic behavior of the stochastic process exhibits interesting dynamics on its own. This can be seen, for example, in the context of front propagation arising asymptotically in branching Brownian motion. In other cases, interesting dynamical behaviour emerges from stochastic processes after rescaling. Examples include the study of rescaled random walks in random environments (which in some settings converge almost surely to a Brownian motion, and in others to jump processes), or the study of two-dimensional lattice models in statistical mechanics (many of which converge, provably or conjecturally, to Schramm-Loewner Evolutions). The goal of this program is to expose graduate and advanced undergraduate students to a variety of such topics of current interest within the study of dynamical stochastic processes.

Friday, May 17, 2019

The 22nd edition of the Colloque panquébécois de l'Institut des sciences mathématiques (ISM) will be held in Montreal from **May 17th to May 19th, 2019**. The goal of this annual conference is to bring together for a week-end the graduate students in mathematics from the universities of Quebec. This year, the conference will be held at **Université de Montréal**.

The participants are invited to give a 20 minutes talk on a mathematical subject of their interest. In addition, four plenary talks will be given by professors. The conference will also feature a talk by the recipient of the Carl Herz prize, awarded by the ISM.

The registration fee for the event is 20$. This includes a **wine and cheese** activity on Friday night, the **saturday evening supper**, as well as breakfasts, lunches and coffee breaks during the whole week-end. Moreover, the first participants to register for the event will receive a promotional ISM mug!

If you have any questions, don't hesitate to contact us.

Thursday, May 16, 2019

**Introduction to birational classification theory in dimension three and higher**

One of the main themes of algebraic geometry is to classify algebraic varieties and to study various geometric properties of each of the interesting classes. Classical theories of curves and surfaces give a beautiful framework of classification theory. Recent developments provide more details in the case of dimension three. We are going to introduce the three-dimensional story and share some expectations for even higher dimensions.

**Date / Time**: Friday, May 16, 2019 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Saturday, May 11, 2019

Have you always dreamed of inventing your own magic tricks? Would you like to know how math can help? The Institut des sciences mathématiques invites you to see magic tricks using cards, knots and mathematics, but definitely not your intuition... it may trick you!

**Saturday, May 11, **12:00 to 2:00**UQAM, ** Président-Kennedy Building

201, ave du Président-Kennedy

Room PK-1630, Montréal

Friday, May 10, 2019

**Quantum Jacobi forms and applications**

Quantum modular forms were defined in 2010 by Zagier; they are somewhat analogous to ordinary modular forms, but they are defined on the rational numbers as opposed to the upper half complex plane, and have modified transformation properties. In 2016, Bringmann and the author defined the notion of a quantum Jacobi form, naturally marrying the concept of a quantum modular form with that of a Jacobi form (the theory of which was developed by Eichler and Zagier in the 1980s). We will discuss these intertwined topics, emphasizing recent developments and applications. In particular, we will discuss applications to combinatorics, topology (torus knots), and representation theory (VOAs).

**Date / Time**: Friday, May 10, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, May 3, 2019

**The stochastic heat equation and KPZ in dimensions three and higher**

The stochastic heat equation and the KPZ equation appear as the macroscopic limits for a large class of probabilistic models, and the study of KPZ, in particular, led to many fascinating developments in probability over the last decade or so, from the regularity structures to integrable probability. We will discuss a small group of recent results on these equations in simple settings, of the PDE flavour, that fall in line with what one may call naive expectations by an applied mathematician.

**Date / Time**: Friday, May 3, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Saturday, April 27, 2019

The ISM will host a half day workshop (in English) for girls in CEGEP who are interested in math and science. It will consist of hands-on talks, a panel discussion, and an inspirational keynote address, all given by women studying or working in math!

The purpose of the event is to encourage young women to continue their study of math and science. The event also aims to help young women form connections that will aid in their transition from CEGEP to college.

The event is open to girls in CEGEP, their parents and teachers, and is free! Lunch, snacks and drinks will be provided! However, we do ask that potential participants register (on the registration page) by Wednesday, April 24, 2019.

Friday, April 26, 2019

**Distinguishing finitely presented groups by their finite quotients**

If G is a finitely generated group, let C(G) denote the set of finite quotients of G. This talk will survey work on the question of to what extent C(G) determines G up to isomorphism, culminating in a discussion of examples of Fuchsian and Kleinian groups that are determined by C(G) (amongst finitely generated residually finite groups).

**Date / Time**: Friday, April 26, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, April 12, 2019

**Linking in torus bundles and Hecke L functions**

Torus bundles over the circle are among the simplest and cutest examples of 3-dimensional manifolds. After presenting some of these examples, using in particular animations realized by Jos Leys, I will consider periodic orbits in these fiber bundles over the circle. We will see that their linking numbers --- that are rational numbers by definition --- can be computed as certain special values of Hecke L-functions. Properly generalized this viewpoint makes it possible to give new topological proof of now classical rationality or integrality theorems of Klingen-Siegel and Deligne-Ribet. It also leads to interesting new "arithmetic lifts" that I will briefly explain. All this is extracted from an on going joint work with Pierre Charollois, Luis Garcia and Akshay Venkatesh.

**Date / Time**: Friday, April 12, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, March 29, 2019

**Principal Bundles in Diophantine Geometry**

Principal bundles and their moduli have been important in various aspects of physics and geometry for many decades. It is perhaps not so well-known that a substantial portion of the original motivation for studying them came from number theory, namely the study of Diophantine equations. I will describe a bit of this history and some recent developments.

**Date / Time**: Friday, March 29, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, March 22, 2019

**Flexibility in contact and symplectic geometry**

We discuss a number of h-principle phenomena which were recently discovered in the field of contact and symplectic geometry. In generality, an h-principle is a method for constructing global solutions to underdetermined PDEs on manifolds by systematically localizing boundary conditions. In symplectic and contact geometry, these strategies typically are well suited for general constructions and partial classifications. Some of the results we discuss are the characterization of smooth manifolds admitting contact structures, high dimensional overtwistedness, the symplectic classification of flexibile Stein manifolds, and the construction of exotic Lagrangians in C^{n}.

**Date / Time**: Friday, March 22, 2019 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Friday, March 15, 2019

Le Sommet de la statistique à Montréal (SÉSÀM) 2019 est un événement qui réunira au cours d’une journée d’activités et de séminaires les étudiants en statistique des universités québécoises.

Avec des conférences plénières, des séminaires étudiants et un concours de conférences-éclairs, le SÉSÀM offrira une occasion privilégiée pour le partage et la discussion entre étudiant-es et avec des chercheurs de renom.

Un dîner et un cocktail en soirée vous permettront d'échanger et d'élargir votre réseau, le tout dans une ambiance chaleureuse et décontractée.

Le 15 mars 2019, c'est un rendez-vous à ne pas manquer!

Friday, March 15, 2019

**Persistent homology as an invariant, rather than as an approximation**

Persistent homology is a very simple idea that was initially introduced as a way of understanding the underlying structure of an object from, perhaps noisy, samples of the object, and has been used as a tool in biology, material sciences, mapping and elsewhere. I will try to explain some of this, but perhaps also some more mathematical applications within geometric group theory. Then I'd like to pivot and study the part that traditionally has been thrown away, and show that this piece is relevant to approximation theory (a la Chebyshev), closed geodesics (a la Gromov), and to problems of quantitative topology (joint work with Ferry, Chambers, Dotter, and Manin).

**Date / Time**: Friday, March 15, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, February 22, 2019

**Quantum Modularity in Topology in Physics**

I will discuss a set of inter-related phenomena in topology, physics, and number theory. A natural question in topology is the construction of homological invariants of 3-manifolds. These turn out to be related to certain special 3- dimensional quantum field theories in physics. Both of these scenarios exhibit a fascinating number theoretic phenomenon: quantum modularity. Quantum modular forms, introduced by Zagier, are functions defined only at rational numbers, and in the most general cases are neither analytic nor modular. It is still an open question to develop a general theory which encompasses their behavior. I will overview these relations and discuss recent advances which may shed light on some of these questions.

**Date / Time**: Friday, February 22, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, February 15, 2019

**Discrete subgroups of Lie groups and geometric structure**

Discrete subgroups of Lie groups play a fundamental role in several areas of mathematics. Discrete subgroups of SL(2,R) are well understood, and classified by the geometry of the corresponding hyperbolic surfaces. On the other hand, discrete subgroups of SL(n,R) for n>2, beyond lattices, remain quite mysterious. While lattices in this setting are rigid, there also exist more flexible "thinner" discrete subgroups, which may have large and interesting deformation spaces (some of them with topological and geometric analogies to the Teichmüller space of a surface, giving rise to so-called "higher Teichmüller theory"). We will survey recent progress in constructing and understanding such discrete subgroups from a geometric and dynamical viewpoint.

**Date / Time**: Friday, February 15, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 Sherbrooke Street West, room 1104

Friday, February 8, 2019

**Periodic orbits of Hamiltonian systems: the Conley conjecture and beyond**

One distinguishing feature of Hamiltonian dynamical systems is that such systems, with very few exceptions, tend to have numerous periodic orbits and these orbits carry a lot of information about the dynamics of the system. In 1984 Conley conjectured that a Hamiltonian diffeomorphism (i.e., the time-one map of a Hamiltonian flow) of a torus has infinitely many periodic points. This conjecture was proved by Hingston some twenty years later, in 2004. Similar results for Hamiltonian diffeomorphisms of surfaces of positive genus were also established by Franks and Handel. Of course, one can expect the Conley conjecture to hold for a much broader class of closed symplectic manifolds and this is indeed the case as has been proved by Gurel, Hein and the speaker. However, the conjecture is known to fail for some, even very simple, phase spaces such as the sphere. These spaces admit Hamiltonian diffeomorphisms with finitely many periodic orbits -- the so-called pseudo-rotations -- which are of particular interest in dynamics. In this talk, mainly based on joint work with Gurel, we will discuss the role of periodic orbits in Hamiltonian dynamics and the methods used to prove their existence, and examine the situations where the Conley conjecture does not hold.

**Date / Time**: Friday, February 8, 2019 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Friday, February 1, 2019

**How does the brain works?**

We will go over the basic challenges for understanding the human brain, the role if mathematics and three examples , one in theoretical neuroscience and two in neuro medicine.

**Date / Time**: Friday, February 1, 2019 - 16:00 ** **

**Venue**: McGill University, Burnside Hall , 805 O., rue Sherbrooke, room 1104

Friday, January 25, 2019

**Conférence Nirenberg du CRM en analyse géométrique: Stochastic diffusive behavior at Kirkwood gaps**

One of the well known indications of instability in the Solar system is the presence of Kirkwood gaps in the Asteroid belt. The gaps correspond to resonance between their periods and the period of Jupiter. The most famous ones are period ratios 3:1, 5:2, 7:3. In the 1980s, J. Wisdom and, independently, A. Neishtadt discovered one mechanism of creation for the 3:1 Kirkwood gap. We propose another mechanism of instabilities, based on an a priori chaotic underlying dynamical structure. As an indication of chaos at the Kirkwood gaps, we show that the eccentricity of Asteroids behaves like a stochastic diffusion process. Along with the famous KAM theory this shows a mixed behavior at the Kirkwood gaps: regular and stochastic. This is a joint work with M. Guardia, P. Martin and P. Roldan.

**Date / Time**: Friday, January 25, 2019 - 16:00

** Venue**: CRM, Université de Montréal, Pavillon André-Aisenstadt, room 6254

Friday, January 18, 2019

**The boundary of the Kähler cone**

On any compact Kähler manifold the space of cohomology classes of all possible Kähler forms is an open convex cone inside a finite-dimensional vector space. I will discuss some recent advances in understanding the geometric and analytic properties of the classes on the boundary of this cone, and describe several applications.

**Date / Time**: Friday, January 18, 2019 - 16:00

** Venue**: McGill University, Burnside Hall , 805 O., rue Sherbrooke, room 1104

Friday, January 11, 2019

Seminars in Undergraduate Mathematics in Montreal (SUMM) is an annual event organized by undergraduate mathematics students from Montreal’s four universities.

The 2019 edition of SUMM will be held at the University of Montreal on January, 11, 12 and 13.

Friday, January 11, 2019

**The mathematics of neutron transport**

We discuss the evolving mathematical view of the Neutron Transport Equation (NTE), which describes the flux of neutrons through inhomogeneous fissile materials. Neutron transport theory emerges in the rigorous mathematical literature in the mid-1950s. Its treatment as an integro-differential equation eventually settled in the applied mathematics literature through the theory of c_0-semigroup theory, thanks to the work of Robert Dautray, Louis Lions and collaborators. This paved the way for its spectral analysis which has played an important role in the design of nuclear reactors and nuclear medical equipment. We also look at the natural probabilistic approach to the NTE which has largely been left behind. Connections with methods of branching particle systems, quasi-stationarity for Markov processes and stochastic analysis all lead new ways of characterising solutions and spectral behaviour the NTE. In particular this, in turn, leads to the suggestion of completely new Monte-Carlo algorithms, which has genuine industrial impact.

**Date / Time**: Friday, January 11, 2019 - 16:00

** Venue**: McGill University, Burnside Hall , 805 O., rue Sherbrooke, room 1104

Friday, December 7, 2018

**Stability problems in general relativity**

There exist several remarkable explicit solutions of Einstein's field equations of General Relativity. A fundamental problem (with implications even for experimental science) is to determine their properties upon perturbation of their initial conditions. I will describe two such solutions: Minkowski spacetime, which is a model for regions of the universe without matter or energy content; and the Kerr--de Sitter family of spacetimes describing (rotating) black holes. In recent work, in parts joint with A. Vasy, we prove global existence and obtain a precise asymptotic description of perturbations of these spacetimes. I will explain these results and indicate the role played by modern microlocal and spectral theoretic techniques in our proofs.

**Date / Time**: Friday, December 7, 2018 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Friday, November 30, 2018

**Completeness of the isomorphism problem for separable C*-algebras**

In logic and computer science one often studies the complexity of decision problems. In mathematical logic this leads to the program of study of relative complexity of isomorphism problems and determining various complexity classes. Broadly speaking, a problem p in a class C is complete in C if any other problem in C reduces to p. The isomorphism problem for separable C*-algebras has been studied since the 1960's and evolved into the Elliott program that classifies C*-algebras via their K-theoretic invariants. During the talk I will discuss the complexity of the isomorphism problem for separable C*-algebras and its completeness in the class of orbit equivalence relations.

**Date / Time**: Friday, November 30, 2018 - 16:00

** Venue**: McGill University, Burnside Hall , 805 O., rue Sherbrooke, room 1104

Friday, November 16, 2018

**Sharp arithmetic transitions for 1D quasiperiodic operators**

A very captivating question in solid state physics is to determine/understand the hierarchical structure of spectral features of operators describing 2D Bloch electrons in perpendicular magnetic fields, as related to the continued fraction expansion of the magnetic flux. In particular, the hierarchical behavior of the eigenfunctions of the almost Mathieu operators, despite significant numerical studies and even a discovery of Bethe Ansatz solutions has remained an important open challenge even at the physics level. I will present a complete solution of this problem in the exponential sense throughout the entire localization regime. Namely, I will describe the continued fraction driven hierarchy of local maxima, and a universal (also continued fraction expansion dependent) function that determines local behavior of all eigenfunctions around each maximum, thus giving a complete and precise description of the hierarchical structure. In the regime of Diophantine frequencies and phase resonances there is another universal function that governs the behavior around the local maxima, and a reflective-hierarchical structure of those, phenomena not even described in the physics literature. These results lead also to the proof of sharp arithmetic transitions between pure point and singular continuous spectrum, in both frequency and phase, as conjectured since 1994. This part of the talk is based on the papers joint with W. Liu. Within the singular continuous regime, it is natural to look for further, dimensional transitions. I will present a sharp arithmetic transition result in this regard that holds for the entire class of analytic quasiperiodic potentials, based on the joint work with S. Zhang.

**Date / Time**: Friday, November 16, 2018 - 16:00

** Venue**: CRM, Université de Montréal, Pavillon André-Aisenstadt, room 1140

Friday, November 9, 2018

**Period mappings and Diophantine equations**

I will give some friendly examples introducing the period mapping. This is an analytic mapping which controls many aspects of how algebraic varieties change in families. After that I will explain joint work with Brian Lawrence which shows that one can exploit transcendence properties of the period mapping to prove results about Diophantine equations. For example we give another proof of the Mordell conjecture (originally proved by Faltings): there are only finitely many rational points on an algebraic curve over Q whose genus is at least 2.

**Date / Time**: Friday, November 9, 2018 - 16:00

** Venue**: McGill University, Burnside Hall , 805 O., rue Sherbrooke ATTENTION - SALLE 1B45 - ATTENTION

Friday, November 2, 2018

**The complexity of detecting cliques and cycles in random graphs**

A strong form of the P ≠ NP conjecture holds that no algorithm faster than n^{O(k)} solves the k-clique problem with high probability when the input is an Erdoös–Rényi random graph with an appropriate edge density. Toward this conjecture, I will describe a line of work lower-bounding the average-case complexity of k-clique (and other subgraph isomorphism problems) in weak models of computation: namely, restricted classes of boolean circuits and formulas. Along the way I will discuss some of the history and current frontiers in Circuit Complexity.Joint work with Kenichi Kawarabayashi, Yuan Li and Alexander Razborov.

Joint work with Ken-ichi Kawarabayashi, Yuan Li and Alexander Razborov.

**Date / Time**: Friday, November 2, 2018 - 16:00

** Venue**: CRM, Université de Montréal, Pavillon André-Aisenstadt, room 1355

Friday, October 26, 2018

**A generalized detailed balance relation**

The transition probabilities of reactions J -> K and K -> J in a thermal bath are related because of the time reversal symmetry of fundamental physical laws. The relation is known as detailed balance relation. We study the problems that arise in obtaining a rigorous proof of detailed balance, using deterministic rather than Markovian dynamics. J. Englands’ biological applications of detailed balance are briefly considered.

**Date / Time**: Friday, October 26, 2018 - 16:00

** Venue**: CRM, Université de Montréal, Pavillon André-Aisenstadt, room 6254

Friday, October 19, 2018

**Vacuum Energy of the Universe and nontrivial topological sectors in Quantum Field Theory**

I discuss a new scenario for early cosmology when the inflationary de Sitter phase emerges dynamically. This genuine quantum effect occurs as a result of dynamics of the topologically nontrivial sectors in a strongly coupled non-abelian gauge theory in an expanding universe. I argue that the key element for this idea to work is the presence of nontrivial holonomy in strongly coupled gauge theories. The effect is global in nature, non-analytical in coupling constant, and cannot be formulated in terms of a gradient expansion in an effective local field theory.

I explain the basic ideas of this framework using a simplified 2D quantum field theory where precise computations can be carried out in theoretically controllable way. I move on to generalize the computations to 4D non-abelian gauge field theories. The last (and most important for cosmological applications) part of my talk is based on recent paper [arXiv:1709.09671].

**Date / Time**: Friday, October 19, 2018 - 16:00

** Venue**: CRM, Université de Montréal, Pavillon André-Aisenstadt, room 1140

Friday, October 12, 2018

**Robust estimation in the presence of influential units for skewed finite and infinite populations**

Many variables encountered in practice (e.g., economic variables) have skewed distributions. The latter provide a conducive ground for the presence of influential observations, which are those that have a drastic impact on the estimates if they were to be excluded from the sample. We examine the problem of influential observations in a classical statistic setting as well as in a finite population setting that includes two main frameworks: the design-based framework and the model-based framework. Within each setting, classical estimators may be highly unstable in the presence of influential units. We propose a robust estimator of the population mean based on the concept of conditional bias of a unit, which is a measure of influence. The idea is to reduce the impact of the sample units that have a large conditional bias. The proposed estimator depends on a cut-off value. We suggest selecting the cut-off value that minimizes the maximum absolute estimated conditional bias with respect to the robust estimator. The properties of the proposed estimator will be discussed. Finally, the results of a simulation study comparing the performance of several estimators in terms of bias and mean square error will be presented.

**Date / Time**: Friday, October 12, 2018 - 16:00

** Venue**: CRM, Université de Montréal, Pavillon André-Aisenstadt, room 6254

Friday, October 5, 2018

**Counting lattice walks confined to cones**

The study of lattice walks confined to cones is a very lively topic in combinatorics and in probability theory, which has witnessed rich developments in the past 20 years. In a typical problem, one is given a finite set of allowed steps S in Z^{d}, and a cone C in R^{d}. Clearly, there are |S|^{n} walks of length n that start from the origin and take their steps in S. But how many of them remain the the cone C?

One of the motivations for studying such questions is that lattice walks are ubiquitous in various mathematical fields, where they encode important classes of objects: in discrete mathematics (permutations, trees, words...), in statistical physics (polymers...), in probability theory (urns, branching processes, systems of queues), among other fields.

The systematic study of these counting problems started about 20 years ago. Beforehand, only sporadic cases had been solved, with the exception of walks with small steps confined to a Weyl chamber, for which a general reflection principle had been developed. Since then, several approaches have been combined to understand how the choice of the steps and of the cone influence the nature of the counting sequence a(n), or of the the associated series A(t)=\sum a(n) t^{n}. For instance, if C is the first quadrant of the plane and S only consists of "small" steps, it is now understood when A(t) is rational, algebraic, or when it satisfies a linear, or a non-linear, differential equation. Even in this simple case, the classification involves tools coming from an attractive variety of fields: algebra on formal power series, complex analysis, computer algebra, differential Galois theory, to cite just a few. And much remains to be done, for other cones and sets of steps.

This talk will survey these recent developments, and conclude a series of talks by the author, in the framework of the Aisenstadt chair.

**Date / Time**: Friday, October 5, 2018 - 16:00

** Venue**: CRM, Université de Montréal, Pavillon André-Aisenstadt, room 6254

Friday, September 28, 2018

**A delay differential equation with a solution whose shortened segments are dense**

Simple-looking autonomous delay differential equations *x'(t*)=*f(x(t-r*)) with a real function *f* and single time lag *r*>0 can generate complicated (chaotic) solution behaviour, depending on the shape of *f*. The same could be shown for equations with a variable, state-dependent delay *r=d(x*_{t}), even for the linear case *f(*\xi)=-\alpha\,\xi with alpha>0. Here the argument x_{t} of the *delay functional* *d* is the history of the solution *x* between *t-r* and *t* defined as the function *x*_{t}:[-r,0] to\mathbb{R}) given by *x _{t}(s)=x(t+s)*. So the delay alone may be responsible for complicated solution behaviour. In both cases the complicated behaviour which could be established occurs in a thin dust-like invariant subset of the infinite-dimensional Banach space or manifold of functions [-

**Date / Time**: Friday, September 28, 2018 - 16:00

** Venue**: McGill University, Burnside Hall , 805 W., rue Sherbrooke, room 1104

Friday, September 21, 2018

**Algebraic structures for topological summaries of data**

This talk introduces an algebraic framework to encode, compute, and analyze topological summaries of data. The main motivating problem, from evolutionary biology, involves statistics on a dataset comprising images of fruit fly wing veins, which amount to embedded planar graphs with varying combinatorics. Additional motivation comes from statistics more generally, the goal being to summarize unknown probability distributions from samples. The algebraic structures for topological summaries take their cue from graded polynomial rings and their modules, but the theory is complicated by the passage from integer exponent vectors to real exponent vectors. The key to making the structures practical for data science applications is a finiteness condition that encodes topological tameness -- which occurs in all modules arising from data -- robustly, in equivalent combinatorial and homological algebraic ways. Out of the tameness condition surprisingly falls much of ordinary commutative algebra, including syzygy theorems and primary decomposition.

**Date / Time**: Friday, September 21, 2018 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Friday, September 14, 2018

**Systems of points with Coulomb interactions**

Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and they give rise to a variety of questions pertaining to analysis, Partial Differential Equations and probability. We will first review these motivations, then present the "mean-field" derivation of effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order behavior, giving information on the configurations at the microscopic level and connecting with crystallization questions, and finish with the description of the effect of temperature.

**Date / Time**: Friday, September 14, 2018 - 16:00

** Venue**: CRM, Université de Montréal, Pavillon André-Aisenstadt, room 6254

Friday, September 7, 2018

**Mathematical challenges in constructing quantum field theory models**

This talk is an overview of algebraic quantum field theory (AQFT) and its perturbative generalization: pAQFT. Both are axiomatic systems meant to provide foundations for quantum field theory (the theory underlying particle physics). I will explain what is the current status of constructing physically relevant models in both approaches and present future perspectives. The most recent results include applications of pAQFT in Yang-Mills theories and effective quantum gravity, as well as some progress in understanding how to go beyond the perturbation theory.

**Date / Time**: Friday, September 7, 2018 - 16:00

** Venue**: CRM, Université de Montréal, Pavillon André-Aisenstadt, room 6254

Sunday, June 10, 2018

**Le carréousel du géomètre**

On the theme of movement, the ISM invites you to participate in animated activities for all ages. A hodge podge of mathematical curiosities will debunk your preconceived notions about all things that roll. Square wheels, carefully constructed manholes, and drills to make square holes are just a few of the objects with which you'll be able to experiment.

**Meet us at the Eureka Festival in Montreal's Old Port**

Conceived and animated by: Alexis Langlois-Rémillard, Olivier Binette and Pierre-Alexandre Mailhot

Monday, May 28, 2018

The summer school will be held at McGill University in Montreal, May 28-June 01, 2018. This school immediately precedes the Statistical Society of Canada Annual Meeting, also to be held at McGill University, which will in turn be followed by a month-long program at the Centre de recherche mathematique on “Causal inference in the presence of dependence and network structure: modelling strategies and model selection”.

Friday, May 25, 2018

The Université de Sherbrooke’s department of mathematics invites you to be a part of the 21st edition of the colloque panquébécois des étudiants et étudiantes de l’ISM which will be held from may 25th to may 27th this year.

This year, four researchers will give a plenary talk to present a part of their research. Louigi Addario-Berry from the Université McGill, Karim Oualkacha from the Université du Québec à Montréal, Hugo Chapdelaine from the Université Laval and Vasilisa Schramchenko from the Université de Sherbrooke will share the auditorium throughout the weekend.

We also invite you to share your own research as part of one of the 20 minutes student’s talks.

A beautiful city, a beautiful campus, great food and amazing people are on schedule for this weekend !

You can find the registration form, the event’s details and how to join us on our website: https://www.ism21colloque.com.

Saturday, May 12, 2018

Have you always dreamed of inventing your own magic tricks? Would you like to know how math can help? The Institut des sciences mathématiques invites you to see magic tricks using cards, knots and mathematics, but definitely not your intuition... it may trick you!

**Saturday, May 12, **11:00 am to 2:00 pm**UQAM, ** Président-Kennedy Building

201, ave du Président-Kennedy

Room PK-1630, Montréal

Friday, May 4, 2018

**Klein-Gordon-Maxwell-Proca systems in the Riemannian setting**

We intend to give a general talk about Klein-Gordon-Maxwell-Proca systems which we aim to be accessible to a broad audience. We will insist on the Proca contribution and then discuss the kind of results one can prove in the electro-magneto static case of the equations.

**Date / Time**: Friday, May 4, 2018 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Friday, April 13, 2018

**Local-global principles in number theory**

One of the classical tools of number theory is the so-called local-global principle, or Hasse principle, going back to Hasse's work in the 1920's. His first results concern quadratic forms, and norms of number fields. Over the years, many positive and negative results were proved, and there is nowa huge number of results in this topic.

This talk will present some old and new results, in particular in the continuation of Hasse's cyclic norm theorem. These have been obtained jointly with Parimala and Tingyu Lee.

**Date / Time**: Friday, April 13, 2018 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Thursday, March 1, 2018

**p-Adic Variation in the Theory of Automorphic Forms**

This will be an expository lecture intended to illustrate through examples the theme of p-adic variation in the classical theory of modular forms. Classically, modular forms are complex analytic objects, but because their fourier coefficients are typically integral, it is possible to do elementary arithmetic with them. Early examples arose already in the work of Ramanujan. Today one knows that modular forms encode deep arithmetic information about elliptic curves and galois representations. The main goal of the lecture will be to motivate a beautiful theorem of Robert Coleman and Barry Mazur, who constructed the so-called Eigenvariety, which leads to a geometric approach to varying modular forms, their associated galois representations, as well as their L-functions, in p-adic analytic families. We will briefly discuss important applications to Number Theory and Iwasawa Theory.

**Date / Time**: Thursday, March 1, 2018 - 15:30

** Venue**: Université Laval, room 3840, Alexandre-Vachon Building

Friday, February 23, 2018

**Cluster theory of the coherent Satake category**

The affine Grassmannian, though a somewhat esoteric looking object at first sight, is a fundamental algebro-geometric construction lying at the heart of a series of ideas connecting number theory (and the Langlands program) to geometric representation theory, low dimensional topology and mathematical physics.

Historically it is popular to study the category of constructible perverse sheaves on the affine Grassmannian. This leads to the *constructible* Satake category and the celebrated (geometric) Satake equivalence.

More recently it has become apparent that it makes sense to also study the category of perverse *coherent* sheaves (the coherent Satake category). Motivated by certain ideas in mathematical physics this category is conjecturally governed by a cluster algebra structure.

We will illustrate the geometry of the affine Grassmannian in an elementary way, discuss what we mean by a cluster algebra structure and then describe a solution to this conjecture in the case of general linear groups.

**Date / Time**: Friday, February 23, 2018 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Friday, February 16, 2018

**Quantum n-body problem: generalized Euler coordinates (from J-L Lagrange to Figure Eight by Moore and Ter-Martirosyan, then and today)**

The potential of the *n*-body problem, both classical and quantum, depends only on the relative (mutual) distances between bodies. By generalized Euler coordinates we mean relative distances and angles. Their advantage over Jacobi coordinates is emphasized.

The NEW IDEA is to study trajectories in both classical, and eigenstates in quantum systems which depends on relative distances ALONE.

We show how this study is equivalent to the study of

(i) the motion of a particle (quantum or classical) in curved space of dimension *n*(*n*-1)/2

or the study of

(ii) the Euler-Arnold (quantum or classical) - *sl*(*n*(*n*-1)/2, R) algebra top.

The curved space of (i) has a number of remarkable properties. In the 3-body case the *de-Quantization* of quantum Hamiltonian leads to a classical Hamiltonian which solves a ~250-years old problem posed by Lagrange on 3-body planar motion.

**Date:** Friday, February 16, 2018

**Time:** 4:00 p.m.**Place:** UdeM, Pavillon André-Aisenstadt, salle 6254

Friday, February 16, 2018

**The Law of Large Populations: The return of the long-ignored N and how it can affect our 2020 vision**

For over a century now, we statisticians have successfully convinced ourselves and almost everyone else, that in statistical inference the size of the population N can be ignored, especially when it is large. Instead, we focused on the size of the sample, n, the key driving force for both the Law of Large Numbers and the Central Limit Theorem. We were thus taught that the statistical error (standard error) goes down with n typically at the rate of 1/√n. However, all these rely on the presumption that our data have perfect quality, in the sense of being equivalent to a probabilistic sample. A largely overlooked statistical identity, a potential counterpart to the Euler identity in mathematics, reveals a Law of Large Populations (LLP), a law that we should be all afraid of. That is, once we lose control over data quality, the systematic error (bias) in the usual estimators, relative to the benchmarking standard error from simple random sampling, goes up with N at the rate of √N. The coefficient in front of √N can be viewed as a data defect index, which is the simple Pearson correlation between the reporting/recording indicator and the value reported/recorded. Because of the multiplier√N, a seemingly tiny correlation, say, 0.005, can have detrimental effect on the quality of inference. Without understanding of this LLP, “big data” can do more harm than good because of the drastically inflated precision assessment hence a gross overconfidence, setting us up to be caught by surprise when the reality unfolds, as we all experienced during the 2016 US presidential election. Data from Cooperative Congressional Election Study (CCES, conducted by Stephen Ansolabehere, Douglas River and others, and analyzed by Shiro Kuriwaki), are used to estimate the data defect index for the 2016 US election, with the aim to gain a clearer vision for the 2020 election and beyond.

**Date:** Friday, February 16, 2018

**Time:** 3:30 p.m.**Place:** McGill University, OTTO MAASS 217

Friday, February 9, 2018

**Persistence modules in symplectic topology**

In order to resolve Vladimir Arnol'd's famous conjecture from the 1960's, giving lower bounds on the number of fixed points of Hamiltonian diffeomorphisms of a symplectic manifold, Andreas Floer has associated in the late 1980's a homology theory to the Hamiltonian action functional on the loop space of the manifold. It was known for a long time that this homology theory can be filtered by the values of the action functional, yielding information about metric invariants in symplectic topology (Hofer's metric, for example). We discuss a recent marriage between the filtered version of Floer theory and persistent homology, a new field of mathematics that has its origins in data analysis, providing examples of new ensuing results.

**Date / Time**: Friday, February 9, 2018 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Friday, January 12, 2018

Seminars in Undergraduate Mathematics in Montreal (SUMM) is an annual event organized by undergraduate mathematics students from Montreal’s four universities.

Each year, the SUMM committee organizes a bilingual colloquium for undergraduate mathematics students. Our main goal is to bring together students from the universities of Montreal and across Canada, creating a dynamic and stimulating undergraduate mathematical community where they can share ideas and interests.

During the three days of the conference, students are invited to share their interest in an area of mathematics or statistics in a 25, 35 or 45 minute talk. Moreover, four keynote speakers — one professor from each organizing university (Concordia University, McGill University, Université de Montréal and Université du Québec à Montréal), representing different areas of research — are invited to give a 50 minute presentation.

For its ninth edition, SUMM will be held at Concordia University on January 12, 13 and 14. If you are interested in presenting a talk, please indicate so in your registration form. Those giving talks will receive a discount on their ticket price.

Friday, January 12, 2018

**What is quantum chaos?**

Where do eigenfunctions of the Laplacian concentrate as eigenvalues go to infinity? Do they equidistribute or do they concentrate in an uneven way? It turns out that the answer depends on the nature of the geodesic flow. I will discuss various results in the case when the flow is chaotic: the Quantum Ergodicity theorem of Shnirelman, Colin de Verdière, and Zelditch, the Quantum Unique Ergodicity conjecture of Rudnick-Sarnak, the progress on it by Lindenstrauss and Soundararajan, and the entropy bounds of Anantharaman-Nonnenmacher. I will conclude with a recent lower bound on the mass of eigenfunctions obtained with Jin. It relies on a new tool called "fractal uncertainty principle" developed in the works with Bourgain and Zahl.

**Date:** Friday, January 12, 2018

**Time:** 4:00 p.m.**Place:** UdeM, Pavillon André-Aisenstadt, salle 6254

Thursday, December 14, 2017

**The new world of infinite random geometric graphs**

The infinite random or Rado graph R has been of interest to graph theorists, probabilists, and logicians for the last half-century. The graph R has many peculiar properties, such as its categoricity: R is the unique countable graph satisfying certain adjacency properties. Erdös and Rényi proved in 1963 that a countably infinite binomial random graph is isomorphic to R.

Random graph processes giving unique limits are, however, rare. Recent joint work with Jeannette Janssen proved the existence of a family of random geometric graphs with unique limits. These graphs arise in the normed space $\ell^n_\infty$ , which consists of $\mathbb{R}^n$ equipped with the $L_\infty$-norm. Balister, Bollobás, Gunderson, Leader, and Walters used tools from functional analysis to show that these unique limit graphs are deeply tied to the $L_\infty$-norm. Precisely, a random geometric graph on any normed, finite-dimensional space not isometric $\ell^n_\infty$ gives non-isomorphic limits with probability 1.

With Janssen and Anthony Quas, we have discovered unique limits in infinite dimensional settings including sequences spaces and spaces of continuous functions. We survey these newly discovered infinite random geometric graphs and their properties.

**Date:** Thursday, December 14, 2017

**Time:** 3:30 p.m.

**Place:** Université Laval, Pavillon Vachon, room 2830

Friday, December 8, 2017

**Primes with missing digits**

Many famous open questions about primes can be interpreted as questions about the digits of primes in a given base. We will talk about recent work showing there are infinitely many primes with no 7 in their decimal expansion. (And similarly with 7 replaced by any other digit.) This shows the existence of primes in a 'thin' set of numbers (sets which contain at most X^{1-c} elements less than X) which is typically very difficult.

**Date / Time**: Friday, December 8, 2017 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Friday, November 24, 2017

**150 years (and more) of data analysis in Canada**

As Canada celebrates its 150th anniversary, it may be good to reflect on the past and future of data analysis and statistics in this country. In this talk, I will review the Victorian Statistics Movement and its effect in Canada, data analysis by a Montréal physician in the 1850s, a controversy over data analysis in the 1850s and 60s centred in Montréal, John A. MacDonald’s use of statistics, the Canadian insurance industry and the use of statistics, the beginning of mathematical statistics in Canada, the Fisherian revolution, the influence of Fisher, Neyman and Pearson, the computer revolution, and the emergence of data science.

**Date:** Friday, November 24, 2017

**Time:** 3:30 p.m.**Place:** McGill, Leacock Building, room LEA 232

Friday, November 24, 2017

**Complex analysis and 2D statistical physics**

Over the last decades, there was much progress in understanding 2D lattice models of critical phenomena. It started with several theories, developed by physicists. Most notably, Conformal Field Theory led to spectacular predictions for 2D lattice models: e.g., critical percolation cluster a.s. has Hausdorff dimension 91/48, while the number of self-avoiding length *N* walks on the hexagonal lattice grows like (\sqrt{2+\sqrt{2}})^{N} N^{11/32}. While the algebraic framework of CFT is rather solid, rigorous arguments relating it to lattice models were lacking. More recently, mathematical approaches were developed, allowing not only for rigorous proofs of many such results, but also for new physical intuition. We will discuss some of the applications of complex analysis to the study of 2D lattice models.

**Date:** Friday, November 24, 2017

**Time:** 4:00 p.m.**Place:** UdeM, Pavillon André-Aisenstadt, salle 6254

Friday, November 17, 2017

**Recent progress on De Giorgi Conjecture**

Classifying solutions to nonlinear partial differential equations are fundamental research in PDEs. In this talk, I will report recent progress made in classifying some elementary PDEs, starting with the De Giorgi Conjecture (1978). I will discuss the classification of global minimizers and finite Morse index solutions, relation with minimal surfaces and Toda integrable systems, as well as recent exciting developments in fractional De Giorgi Conjecture.

**Date / Time**: Friday, November 17, 2017 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Friday, October 27, 2017

**Beneath the Surface: Geometry Processing at the Intrinsic/Extrinsic Interface**

Algorithms for analyzing 3D surfaces find application in diverse fields from computer animation to medical imaging, manufacturing, and robotics. Reflecting a bias dating back to the early development of differential geometry, a disproportionate fraction of these algorithms focuses on discovering intrinsic shape properties, or those measurable along a surface without considering the surrounding space. This talk will summarize techniques to overcome this bias by developing a geometry processing pipeline that treats intrinsic and extrinsic geometry democratically. We describe theoretically-justified, stable algorithms that can characterize extrinsic shape from surface representations.

In particular, we will show two strategies for computational extrinsic geometry. In our first approach, we will show how the discrete Laplace-Beltrami operator of a triangulated surface accompanied with the same operator for its offset determines the surface embedding up to rigid motion. In the second, we will treat a surface as the boundary of a volume rather than as a thin shell, using the Steklov (Dirichlet-to-Neumann) eigenproblem as the basis for developing volumetric spectral shape analysis algorithms without discretizing the interior.

**Date:** Friday, October 27, 2017

**Time:** 4:00 p.m.**Place:** UdeM, Pavillon André-Aisenstadt, salle 6254

Friday, October 13, 2017

**Supercritical Wave Equations**

I will review the problem of Global existence for dispersive equations, in particular, supercritical equations. These equations who play a fundamental role in science, have been , and remain a major challenge in the field of Partial Differential Equations. They come in various forms, derived from Geometry, General Relativity, Fluid Dynamics, Field Theory. I present a new approach to classify the asymptotic behavior of wave equations, supercritical and others, and construct global solutions with large initial data. I will then describe current extensions to Nonlinear Schroedinger Equations.

**Date:** Friday, October 13, 2017

**Time:** 4:00 p.m.**Place:** UdeM, Pavillon André-Aisenstadt, room 6254

Friday, September 29, 2017

**The first field**

The “first field” is obtained by making the entries in its addition and multiplication tables be the smallest possibilities. It is really an interesting field that contains the integers, but with new addition and multiplication tables. For example, 2 x 2 = 3, 5 x 7 = 13, ... It extends to the infinite ordinals and the first infinite ordinal is the cube root of 2!

**Date:** Friday, September 29, 2017

**Time:** 4:00 p.m.**Place:** UdeM, Pavillon André-Aisenstadt, salle 1140

Friday, September 15, 2017

**Isometric embedding and quasi-local type inequality**

In this talk, we will first review the classic Weyl's embedding problem and its application in quasi-local mass. We will then discuss some recent progress on Weyl's embedding problem in general Riemannian manifold. Assuming isometric embedding into Schwarzschild manifold, we will further establish a quasi-local type inequality. This talk is based on works joint with Pengfei Guan and Pengzi Miao.

**Date / Time**: Friday, September 15, 2017 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Wednesday, July 19, 2017

The CUMC is an academic conference aimed at undergraduate students studying mathematics or mathematical fields. This year's event will take place in Montréal, from the 19th to the 23th of July, in collaboration between Université de Montréal, Concordia University, Université du Québec à Montréal and McGill University.

Monday, June 26, 2017

Concordia’s Math Camp provides an enriching experience for students who have shown an interest or an aptitude for mathematics. Led by a bilingual instructor, experienced in outreach activities and international mathematical contests, and staffed by graduate students in mathematics or mathematics education from the Montreal area universities, the camp will challenge **students aged 10-15** to develop their problem skills while having fun.** **

The students will participate in problem solving sessions, and experience math through games, projects, experiments and other fun activities. The camp will also highlight math in everyday life from practical fields to the arts. Finally, it is a chance for kids interested in mathematics to meet other kids who share the same interest and thus develop new friendships.

**The cost of the camp is $225/week (M-F: 9am to 4pm) with an extra $35/week for extended care (M-F: 8am-9am and 4pm-5pm).** The camp is held in the Mathematics and Statistics Department of Concordia University, 9^{th} floor of the Library Building in SGW campus:

http://www.concordia.ca/maps/sgw-campus.html

A **Montreal Math Circle Activity**, the camp is made possible by **Concordia University** in collaboration with the **Institut des sciences mathématiques (ISM)** and the support of **Canadian Mathematical Society (CMS) **and** NSERC Promoscience**.

Mathematical activities are chosen among:

- appropriate level problem solving;

- challenge problems: like open problems and discovery type activities;

- exploratory activity with math themes like polygonal numbers, math in paintings, Escher logic and tessellations; or a *math walk* around the department.

- recreational mathematics with activities foreseen for outreach and puzzles; stories or movies about mathematicians and mathematics.

We will aim to find at least one daily activity where the students will get to move around and spend some physical energy.

Any general enquiries may be addressed to montrealmathclub@gmail.com. Please write *June camp* in the subject line.

**To register, please complete the ****form****.**

Saturday, May 13, 2017

Have you always dreamed of inventing your own magic tricks? Would you like to know how math can help? The Institut des sciences mathématiques invites you to see magic tricks using cards, knots and mathematics, but definitely not your intution... it may trick you!

**Saturday, May 13, **10:00 am to 2:00 pm**UQAM, ** Président-Kennedy Building

201, ave du Président-Kennedy

Room PK-R650, Montréal

Friday, May 12, 2017

The purpose of this annual conference is to bring together for a weekend graduate students from the province of Quebec enrolled in mathematical sciences programs. This year, the symposium will be held at the Université du Québec à Trois-Rivières from May 12-14, 2017. Everyone is invited to register and to present his or her work on a topic that they are passionate about.

Friday, May 5, 2017

**From the geometry of numbers to Arakelov geometry**

Arakelov geometry is a modern formalism that extends in various directions the geometry of numbers founded by Minkowski in the nineteenth century. The objects of study are arithmetic varieties, namely complex varieties that can be defined by polynomial equations with integer coefficients. The theory exploits the interplay between algebraic geometry and number theory and complex analysis and differential geometry. Recently, the formalism found beautiful and important applications to the so-called Kudla programme and the Colmez conjecture. In the talk, I will first introduce elementary facts in Minkowski's geometry of numbers. This will provide a motivation for the sequel, where I will give my own view of Arakelov geometry, by focusing on toy (but non-trivial) examples of one of the central theorems in the theory, the arithmetic Riemann-Roch theorem mainly due to Bismut, Gillet and Soulé, and generalizations. I hope there will be ingredients to satisfy different tastes, for instance modular forms (arithmetic aspect), analytic torsion (analytic aspect) and Selberg zeta functions (arithmetic, analytic and dynamic aspects).

**Date / Time**: Friday, May 5, 2017 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Friday, April 21, 2017

**Introduction to the Energy Identity for Yang-Mills**

In this talk we give an introduction to the analysis of the Yang-Mills equation in higher dimensions. In particular, when studying sequences of solutions we will study the manner in which blow up can occur, and how this blow up may be understood through the classical notions of the defect measure and bubbles. The energy identity is an explicit conjectural relationship, known to be true in dimension four, relating the energy density of the defect measure at a point to the bubbles which occur at that point, and we will give a brief overview of the recent proof of this result for general stationary Yang Mills in higher dimensions. The work is joint with Daniele Valtorta.

**Date / Time**: Friday, April 21, 2017 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Friday, April 21, 2017

**Date /Time ** : Friday, April 21, 2017 - 15:30

** Venue ** : McGill University, Burnside Hall, 805 Sherbrooke Ouest, room 1205

Friday, April 7, 2017

**Kahler-Einstein metrics**

Kahler-Einstein metrics are of fundamental importance in Kahler geometry, with connections to algebraic geometry, geometric analysis, string theory amongst other fields. Their study has received a great deal of attention recently, culminating in the solution of the Yau-Tian-Donaldson conjecture, characterizing which complex manifolds admit Kahler-Einstein metrics. I will give an overview of the field, including some recent developments.

**Date / Time**: Friday, April 7, 2017 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Thursday, April 6, 2017

**Instrumental variable regression with survival outcomes**

Instrumental variable (IV) methods are popular in non-experimental studies to estimate the causal effects of medical interventions or exposures. These approaches allow for the consistent estimation of such effects even if important confounding factors are unobserved. Despite the increasing use of these methods, there have been few extensions of IV methods to censored data regression problems. We discuss challenges in applying IV structural equational modelling techniques to the proportional hazards model and suggest alternative modelling frameworks. We demonstrate the utility of the accelerated lifetime and additive hazards models for IV analyses with censored data. Assuming linear structural equation models for either the event time or the hazard function, we proposed closed-form, two-stage estimators for the causal effect in the structural models for the failure time outcomes. The asymptotic properties of the estimators are derived and the resulting inferences are shown to perform well in simulation studies and in an application to a data set on the effectiveness of a novel chemotherapeutic agent for colon cancer.

**Date / Time**: Thursday, April 6, 2017 - 15:30

** Venue**: Laval University, Pavillon Vachon, room 3840

Friday, March 31, 2017

**PDEs on non-smooth domains**

Abstract: In these lecture we will discuss the relationship between the boundary regularity of the solutions to elliptic second order divergence form partial differential equations and the geometry of the boundary of the domain where they are defined. While in the smooth setting tools from classical PDEs are used to address this question, in the non-smooth setting techniques from harmonic analysis and geometric measure theory are needed to tackle the problem. The goal is to present an overview of the recent developments in this very active area of research.

**Date / Time**: Friday, March 31, 2017 - 16:00

** Venue**: UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Friday, March 17, 2017

**Inference in Dynamical Systems**

We consider the asymptotic consistency of maximum likelihood parameter estimation for dynamical systems observed with noise. Under suitable conditions on the dynamical systems and the observations, we show that maximum likelihood parameter estimation is consistent. Furthermore, we show how some well-studied properties of dynamical systems imply the general statistical properties related to maximum likelihood estimation. Finally, we exhibit classical families of dynamical systems for which maximum likelihood estimation is consistent. Examples include shifts of finite type with Gibbs measures and Axiom A attractors with SRB measures. We also relate Bayesian inference to the thermodynamic formalism in tracking dynamical systems.

**Date / Time ** : Friday, March 17, 2017 - 15:30

** Venue ** : McGill University, Burnside Hall, 805 Sherbrooke Ouest, salle 1205

Friday, March 10, 2017

**Probabilistic aspects of minimum spanning trees**

One of the most dynamic areas of probability theory is the study of the behaviour of discrete optimization problems on random inputs. My talk will focus on the probabilistic analysis of one of the first and foundational combinatorial optimization problems: the minimum spanning tree problem. The structure of a random minimum spanning tree (MST) of a graph G turns out to be intimately linked to the behaviour of critical and near-critical percolation on G. I will describe this connection, and present some results on the structure, scaling limits, and volume growth of random MSTs. It turns out that, on high-dimensional graphs, random minimum spanning trees are expected to be three-dimensional when viewed intrinsically, and six-dimensional when viewed as embedded objects.

Based on joint works with Nicolas Broutin, Christina Goldschmidt, Simon Griffiths, Ross Kang, Gregory Miermont, Bruce Reed, Sanchayan Sen.

**Date / Time ** : Friday, March 10, 2017 - 4:00 PM

** Venue ** : CRM, Université de Montréal, Pavillon André-Aisenstadt, 2920 Chemin de la Tour, room 6254

Friday, February 24, 2017

**Spreading phenomena in integrodifference equations with overcompensatory growth function**

The globally observed phenomenon of the spread of invasive biological species with all its sometimes detrimental effects on native ecosystems has spurred intense mathematical research and modelling efforts into corresponding phenomena of spreading speeds and travelling waves. The standard modelling framework for such processes is based on reaction- diffusion equations, but several aspects of an invasion can only be appropriately described by a discrete-time analogues, called integrodifference equations. The theory of spreading speeds and travelling waves in such integrodifference equations is well established for the "mono-stable" case, i.e. when the non-spatial dynamics show a globally stable positive steady state. When the positive state of the non-spatial dynamics is not stable, as is the case with the famous discrete logistic equation, it is unclear how the corresponding spatial spread profile evolves and at what speed. Previous simulations seemed to reveal a travelling profile in the form of a two-cycle, with or without spatial oscillations. The existence of a travelling wave solution has been proven, but its shape and stability remain unclear. In this talk, I will show simulations that suggest that there are several travelling profiles at different speeds. I will establish corresponding generalizations of the concept of a spreading speed and prove the existence of such speeds and travelling waves in the second- iterate operator. I conjecture that rather than a travelling two-cycle for the next-generation operator, one observes a pair of stacked fronts for the second-iterate operator. I will relate the observations to the phenomenon of dynamic stabilization.

**Date / Time ** : Friday, February 24, 2017 - 4:00 PM

** Venue ** : CRM, Université de Montréal, Pavillon André-Aisenstadt, 2920 Chemin de la Tour, room 6254

Friday, February 10, 2017

**Knot concordance**

I will introduce the knot concordance group, give a survey of our current understanding of it and discuss some relationships with the topology of 4-manifolds.

**Date / Time**: Friday, February 10, 2017 - 4:00 PM

** Venue ** UQAM, Président-Kennedy Building, 201, ave du Président-Kennedy, room PK-5115

Friday, January 20, 2017

**The Birch-Swinnerton Dyer Conjecture and counting elliptic curves of ranks 0 and 1**

This colloquium talk will begin with an introduction to the Birch--Swinnerton-Dyer conjecture for elliptic curves -- just curves defined by the equations y^{2}=x^{3}+Ax+B -- and then describe recent advances that allow us to prove that lots of elliptic curves have rank zero or one.

**Date / Time**: Friday, January 20, 2017 - 4:00 PM

** Venue ** UQAM, Président-Kennedy Building, 201, ave du Président-Kennedy, room PK-5115

Friday, January 13, 2017

The Seminars in Undergraduate Mathematics in Montreal (SUMM) is an annual event organized by students who are currently enrolled in an undergraduate mathematics program at one of the four Montreal universities.

The 2017 edition SUMM will be held at McGill University on January, 13, 14 and 15.

Friday, December 2, 2016

**Partial differential equations of mixed elliptic-hyperbolic type in mechanics and geometry**

As is well-known, two of the basic types of linear partial differential equations (PDEs) are hyperbolic PDEs and elliptic PDEs, following the classification for linear PDEs first proposed by Jacques Hadamard in the 1920s; and linear theories of PDEs of these two types have been well established, respectively. On the other hand, many nonlinear PDEs arising in mechanics, geometry, and other areas naturally are of mixed elliptic-hyperbolic type. The solution of some longstanding fundamental problems in these areas greatly requires a deep understanding of such nonlinear PDEs of mixed type. Important examples include shock reflection-diffraction problems in fluid mechanics (the Euler equations) and isometric embedding problems in differential geometry (the Gauss-Codazzi-Ricci equations), among many others. In this talk we will present natural connections of nonlinear PDEs of mixed elliptic-hyperbolic type with these longstanding problems and will then discuss some recent developments in the analysis of these nonlinear PDEs through the examples with emphasis on developing and identifying mathematical approaches, ideas, and techniques for dealing with the mixed-type problems. Further trends, perspectives, and open problems in this direction will also be addressed.

**Date / Time**: Friday, December 2, 2016 - 4:00 PM

** Venue ** : UQAM, Président-Kennedy Building, 201, ave du Président-Kennedy, room PK-5115

Thursday, December 1, 2016

**High-dimensional changepoint estimation via sparse projection**

Changepoints are a very common feature of Big Data that arrive in the form of a data stream. We study high-dimensional time series in which, at certain time points, the mean structure changes in a sparse subset of the coordinates. The challenge is to borrow strength across the coordinates in order to detect smaller changes than could be observed in any individual component series. We propose a two-stage procedure called 'inspect' for estimation of the changepoints: first, we argue that a good projection direction can be obtained as the leading left singular vector of the matrix that solves a convex optimisation problem derived from the CUSUM transformation of the time series. We then apply an existing univariate changepoint detection algorithm to the projected series. Our theory provides strong guarantees on both the number of estimated changepoints and the rates of convergence of their locations, and our numerical studies validate its highly competitive empirical performance for a wide range of data generating mechanisms.

**Date / Time ** : Thursday, December 1, 2016 - 15:30

** Venue ** : Room 1205, Burnside Hall, 805 Sherbrooke West

Friday, November 25, 2016

**Around the Möbius function**

The Moebius function plays a central role in number theory; both the prime number theorem and the Riemann Hypothesis are naturally formulated in terms of the amount of cancellations one gets when summing the Moebius function. In recent joint work with K. Matomaki the speaker showed that the sum of the Moebius function exhibits cancellations in "almost all intervals" of increasing length. This goes beyond what was previously known conditionally on the Riemann Hypothesis. The result holds in fact in greater generality. Exploiting this generality one can show that between a fixed number of consecutive squares there is always an integer composed of only "small" prime factors. This is related to the running time of Lenstra's factoring algorithm. I will also discuss some further developments : the work of Tao on correlations between consecutive values of Chowla, and his application of this result to the resolution of the Erdos discrepancy problem.** **

**Date / Time**: Friday, November 25, 2016 - 4:00 PM

** Venue ** UQAM, Président-Kennedy Building, 201, ave du Président-Kennedy, room PK-5115

Friday, November 4, 2016

**The nonlinear stability of Minkowski space for self-gravitating massive fields**

I will review results on the global evolution of self-gravitating massive matter in the context of Einstein's theory as well as the f(R)-theory of gravity. In collaboration with Yue Ma (Xian), I have investigated the global existence problem for the Einstein equations coupled with a Klein-Gordon equation describing the evolution of a massive scalar field. Our main theorem establishes the global nonlinear stability of Minkowski spacetime upon small perturbations of the metric and the matter field. Recall that the fully geometric proof by Christodoulou and Klainerman in 1993, as well as the proof in wave gauge by Lindblad and Rodnianski in 2010, both apply to vacuum spacetimes and massless fields only. Our new technique of proof, which we refer to as the Hyperboloidal Foliation Method, does not use Minkowski's scaling field and is based on a foliation of the spacetime by asymptotically hyperboloidal spacelike hypersurfaces, on sharp estimates for wave and Klein-Gordon equations, and on an analysis of the quasi-null hyperboloidal structure (as we call it) of the Einstein equations in wave gauge.

**Date / Time**: Friday, November 4, 2016 - 4:00 PM

** Venue ** UQAM, Président-Kennedy Building, 201, ave du Président-Kennedy, room PK-5115

Friday, October 28, 2016

**Efficient tests of covariate effects in two-phase failure time studies**

Two-phase studies are frequently used when observations on certain variables are expensive or difficult to obtain. One such situation is when a cohort exists for which certain variables have been measured (phase 1 data); then, a sub-sample of individuals is selected, and additional data are collected on them (phase 2). Efficiency for tests and estimators can be increased by basing the selection of phase 2 individuals on data collected at phase 1. For example, in large cohorts, expensive genomic measurements are often collected at phase 2, with oversampling of persons with “extreme” phenotypic responses. A second example is case-cohort or nested case-control studies involving times to rare events, where phase 2 oversamples persons who have experienced the event by a certain time. In this talk I will describe two-phase studies on failure times, present efficient methods for testing covariate effects. Some extensions to more complex outcomes and areas needing further development will be discussed.

**Date:** Friday, October 28, 2016

**Time:** 3:30 p.m. - 4:30 p.m.

**Place:** Room 1205, Burnside Hall, 805 Sherbrooke West

Friday, October 21, 2016

**Integrable probability and the KPZ universality class**

I will explain how certain integrable structures give rise to meaningful probabilistic systems and methods to analyze them. Asymptotics reveal universal phenomena, such as the Kardar-Parisi-Zhang universality class. No prior knowledge will be assumed.

**Date / Time**: Friday, October 21, 2016 - 4:00 PM

** Venue ** : CRM, André-Aisenstadt Building, 2920 chemin de la tour, room 6254

Friday, October 14, 2016

**Rigorously verified computing for infinite dimensional nonlinear dynamics: a functional analytic approach**

Studying and proving existence of solutions of nonlinear dynamical systems using standard analytic techniques is a challenging problem. In particular, this problem is even more challenging for partial differential equations, variational problems or functional delay equations which are naturally defined on infinite dimensional function spaces. The goal of this talk is to present rigorous numerical technique relying on functional analytic and topological tools to prove existence of steady states, time periodic solutions, traveling waves and connecting orbits for the above mentioned dynamical systems. We will spend some time identifying difficulties of the proposed approach as well as time to identify future directions of research.

**Date / Time**: Friday, October 14, 2016 - 4:00 PM

** Venue ** : CRM, André-Aisenstadt Building, 2920 chemin de la tour, room 6254

Friday, September 30, 2016

**Notions of simplicity in low-dimensions**

Various auxiliary structures arise naturally in low-dimensions. I will discuss three of these: left-orders on the fundamental group, taut foliations on three-manifolds, and non-trivial Floer homological invariants. Perhaps surprisingly, for (closed, connected, orientable, irreducible) three-manifolds, it has been conjectured that the existence of any one of these structures implies the others. I will describe what is currently known about this conjectural relationship, as well as some of the machinery — particularly in Heegaard Floer theory — that has been developed in pursuit of the conjecture.

**Date / Time**: Friday, September 30, 2016 - 4:00 PM

** Venue ** UQAM, Président-Kennedy Building, 201, ave du Président-Kennedy, room PK-5115

Friday, September 16, 2016

**Statistical Inference for fractional diffusion processes**

There are some time series which exhibit long-range dependence as noticed by Hurst in his investigations of river water levels along Nile river. Long-range dependence is connected with the concept of self-similarity in that increments of a self-similar process with stationary increments exhibit long-range dependence under some conditions. Fractional Brownian motion is an example of such a process. We discuss statistical inference for stochastic processes modeled by stochastic differential equations driven by a fractional Brownian motion. These processes are termed as fractional diffusion processes. Since fractional Brownian motion is not a semimartingale, it is not possible to extend the notion of a stochastic integral with respect to a fractional Brownian motion following the ideas of Ito integration. There are other methods of extending integration with respect to a fractional Brownian motion. Suppose a complete path of a fractional diffusion process is observed over a finite time interval. We will present some results on inference problems for such processes.

**Date:** Friday, September 16, 2016

**Time:** 4:00 p.m.**Place:** Concordia University, Library Building, 1400 de Maisonneuve O., room LB-921.04

Friday, September 16, 2016

**Cubature, approximation, and isotropy in the hypercube**

The hypercube is the standard domain for computation in higher dimensions. We describe two respects in which the anisotropy of this domain has practical consequences. The first is a matter well known to experts (and to Chebfun users): the importance of axis-alignment in low-rank compression of multivariate functions.

Rotating a function by a few degrees in two or more dimensions may change its numerical rank completely. The second is new. The standard notion of degree of a multivariate polynomial, total degree, is isotropic – invariant under rotation.

The hypercube, however, is highly anisotropic. We present a theorem showing that as a consequence, the convergence rate of multivariate polynomial approximations in a hypercube is determined not by the total degree but by the *Euclidean degree*, defined in terms of not the 1-norm but the 2-norm of the exponent vector **k** of a monomial x_{1}^{k1}... x_{s}^{ks}. The consequences, which relate to established ideas of cubature and approximation going back to James Clark Maxwell, are exponentially pronounced as the dimension of the hypercube increases. The talk will include numerical demonstrations.

**Date:** Friday, September 16, 2016

**Time:** 4:00 p.m.**Place:** UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Monday, July 4, 2016

The goal of the 2016 CRM Summer School in Quebec City is to prepare students for research involving spectral theory. The school will give an overview of a selection of topics from spectral theory, interpreted in a broad sense. It will cover topics from pure and applied mathematics, each of which will be presented in a 5-hour mini-course by a leading expert. These lectures will be complemented by supervised computer labs and exercise sessions. At the end of the school, invited speakers will give specialized talks. This rich subject intertwines several sub-disciplines of mathematics, and it will be especially beneficial to students. The subject is also very timely, as spectral theory is witnessing major progresses both in its mathematical sub-disciplines and in its applications to technology and science in general.

The school is intended to advanced undergraduate and beginning graduate students. As such, the prerequisites will be kept at a minimum, and review material will be provided a few weeks before the event.

Sunday, June 12, 2016

All are welcome to attend the first annual Mathfest for a morning of mathematical games and discoveries.

**Where:** John Molson Building, Concordia University, room MB 3.430, 1450 rue Guy

**When: **June 12, 10:00 AM - 12:00 noon

FREE

The event is organized by Concordia's Department of Mathematics and Statistics and by the ISM.

Friday, May 20, 2016

**Complexité des fonctions d'un grand nombre de variables: de la physique statistique aux algorithmes de "deep learning"**

**Date / Time:** Friday, May 20, 2016 - 4:00 PM

**Venue:** CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

Friday, May 13, 2016

The goal of this annual conference is to bring together Quebec graduate students in the mathematical sciences for a weekend. This year, the conference will be held at UQAM from May 13-15, 2016. Everyone is invited to come or present their work on a subject they are interested in.

The 20-minute talks given by graduate students are a great chance to learn about the work of your colleagues. They will be grouped in thematic sessions that will cover a variety of subjects such as Combinatorics, Financial Mathematics, Mathematical Physics, etc. In addition, four plenary talks will be given by well-known researchers. Presentations in either French or English are welcome. Since the ISM is celebrating its 25th anniversary this year, all the plenary talks at the conference will be given by former ISM students.

The conference will be launched with a social activity allowing everyone to get to know each other.

To register, click here.

We look forward to seeing you this spring!

Friday, April 15, 2016

**Elliptic PDEs in two dimensions**

I will give a short survey of the several approaches to the regularity theory of elliptic equations in two dimensions. In particular I will focus on some old ideas of Bernstein and their application to the infinity Laplace equation and to the Bellman equation in two dimensions.

**Date / Time ** : Friday, April 15, 2016 - 16:00

** Venue ** : UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Thursday, April 14, 2016

**Statistical Estimation Problems in Meta-Analysis**

The principal statistical estimation problem in meta-analysis is to obtain a reliable confidence interval for the treatment effect. Several possible approaches and settings are described. In particular a Bayesian model with non-informative priors and the default data-dependent priors is discussed along with relevant optimization issues.

**Date:** Thursday, April 14, 2016

**Time:** 4:30 p.m

**Venue:** Université de Sherbrooke, 2500, boul. de l'Université, salle D3-2041

Thursday, April 14, 2016

**The statistical price for computational efficiency**

With the explosion of the size of data, computation has become an integral part of statistics. Ad hoc remedies such as employing convex relaxations, or manipulating sufficient statistics, have been successful to derive efficient procedures with provably optimal statistical guarantees. Unfortunately, computational efficiency sometimes comes at an inevitable statistical cost. Therefore, one needs to redefine optimality among computationally efficient procedures. Using tools from information theory and computational complexity, we quantify this cost in the context of two models: (i) the multi-armed bandit problem, and (ii) sparse principal component analysis [Based on joint work with Q. Berthet, S. Chassang, V. Perchet and E. Snowberg]

**Date /Time ** : Thursday, April 14, 2016 - 15:30

** Venue ** : Laval University, Pavillon Adrien-Pouliot, room 2840

Friday, April 8, 2016

**The dimer model: universality and conformal invariance**

The dimer model on a finite bipartite planar graph is a uniformly chosen set of edges which cover every vertex exactly once. It is a classical model of statistical mechanics, going back to work of Kasteleyn and Temperley/Fisher in the 1960s who computed its partition function.

After giving an overview, I will discuss some recent joint work with Benoit Laslier and Gourab Ray, where we prove in a variety of situations that when the mesh size tends to 0 the fluctuations are described by a universal and conformally invariant limit known as the Gaussian free field.

A key novelty in our approach is that the exact solvability of the model plays only a minor role. Instead, we rely on a connection to imaginary geometry, where Schramm-Loewner Evolution curves are viewed as flow lines of an underlying Gaussian free field.

**Date / Time ** : Friday, April 8, 2016 - 16:00

** Venue ** : CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, room 6214

Friday, April 1, 2016

**Needles, Bushes, Hairbrushes and Polynomials**

Pretend that your car is a unit line segment. How do you perform a three point turn using an infinitesimally small area on the road? It turns out that this seemingly impossible driving stunt is related to the fundamental theorem of calculus, as well as all the objects in the title of this talk! We will explore these connections and see how they have been useful in many problems in mathematics.

**Date / Time ** : Friday, April 1, 2016 - 16:00

** Lieu/Venue ** : UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Friday, March 18, 2016

**Harry Potter's Cloak via Transformation Optics**

Can we make objects invisible? This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc., including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter. In the last decade or so, there have been several scientific proposals to achieve invisibility. We will introduce some of these in a non-technical fashion, concentrating on the so-called "transformation optics" that has received the most attention in the scientific literature.

**Date / Time ** : Friday, March 18, 2016 - 16:00

** Venue ** : CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, room 6214

Thursday, March 17, 2016

**Quantum Chromatic Numbers and the conjectures of Connes and Tsirelson**

It is possible to characterize the chromatic number of a graph in terms of a game. It is the fewest number of colours for which a winning strategy exists using classical random variables to a certain graph colouring game. If one allows the players to use quantum experiments to generate their random outcomes, then for many graphs this game can be won with far fewer colours. This leads to the definition of the quantum chromatic number of a graph. However, there are several mathematical models for the set of probability densities generated by quantum experiments and whether or not these models agree depends on deep conjectures of Connes and Tsirelson. Thus, there are potentially several "different" quantum chromatic numbers and computing them for various graphs gives us a combinatorial means to test these conjectures. In this talk I will present these ideas and some of the results in this area. I will only assume that the audience is familiar with the basics of Hilbert space theory and assume no background in quantum theory.

**Date / Time ** : Thursday, March 17, 2016 - 15:30

** Venue ** : Laval University, Pavillon Alexandre Vachon, room VCH-2830

Thursday, March 10, 2016

**Ridges and valleys in the high excursion sets of Gaussian random fields**

It is well known that normal random variables do not like taking large values. Therefore, a continuous Gaussian random field on a compact set does not like exceeding a large level. If it does exceed a large level at some point, it tends to go back below the level a short distance away from that point. One, therefore, does not expect the excursion set above a high for such a field to possess any interesting structure. Nonetheless, if we want to know how likely are two points in such an excursion set to be connected by a path ("a ridge") in the excursion set, how do we figure that out? If we know that a ridge in the excursion set exists (e.g. the field is above a high level on the surface of a sphere), how likely is there to be also a valley (e.g. the field going to below a fraction of the level somewhere inside that sphere)?

We use the large deviation approach. Some surprising results (and pictures) are obtained.

**Date / Time ** : Thursday, March 10, 2016 - 15:30

** Venue ** : McGill University, Burnside Hall, salle à venir

Friday, February 26, 2016

**The fundamental theorem of algebra, complex analysis and ... astrophysics**

The fundamental theorem of algebra, complex analysis and ... astrophysicsThe Fundamental Theorem of Algebra first rigorously proved by Gauss states that each complex polynomial of degree $n$ has precisely $n$ complex roots. In recent years various extensions of this celebrated result have been considered. We shall discuss the extension of the FTA to harmonic polynomials of degree $n$. In particular, the theorem of D. Khavinson and G. Swiatek that shows that the harmonic polynomial \bar{z}-p(z), deg \, p=n>1 has at most 3*n*-2 zeros as was conjectured in the early 90's by T. Sheil-Small and A. Wilmshurst. L. Geyer was able to show that the result is sharp for all *n*. G. Neumann and D. Khavinson proved that the maximal number of zeros of rational harmonic functions \bar{z}-r(z), deg \,r =n>1 is 5*n*-5. It turned out that this result confirmed several consecutive conjectures made by astrophysicists S. Mao, A. Petters, H. Witt and, in its final form, the conjecture of S. H. Rhie that were dealing with the estimate of the maximal number of images of a star if the light from it is deflected by *n* co-planar masses. The first non-trivial case of one mass was already investigated by A. Einstein around 1912. We shall also discuss the problem of gravitational lensing of a point source of light, e.g., a star, by an elliptic galaxy, more precisely the problem of the maximal number of images that one can observe. Under some more or less "natural" assumptions on the mass distribution within the galaxy one can prove (A.Eremenko and W. Bergweiler - 2010, also, K - E. Lundberg - 2010) that the number of visible images can never be more than four in some cases and six in the other. Interestingly, the former situation can actually occur and has been observed by astronomers. Still there are much more open questions than there are answers.

**Time ** : Friday, February 26, 2016 - 16:00

** Venue ** : CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, room 6214

Friday, February 12, 2016

**Date / Time ** : February 12, 2016 - 16:00

** Venue ** : CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, room 6214

Thursday, February 11, 2016

AUTHOR:

Michael Harris (Université Paris-Diderot, Columbia University)

MATHEMATICS WITHOUT APOLOGIES

An unapologetic guided tour of the mathematical life

** You may purchase the book on site for $25 (cash only) **

What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers—for the sake of truth, beauty, and practical applications—this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources.

Drawing on his personal experiences and obsessions as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyám to such contemporary giants as Alexander Grothendieck and Robert Langlands, Michael Harris reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, he touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party?

Disarmingly candid, relentlessly intelligent, and richly entertaining, Mathematics without Apologies takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.

Michael Harris is professor of mathematics at the Université Paris Diderot and Columbia University. He is the author or coauthor of more than seventy mathematical books and articles, and has received a number of prizes, including the Clay Research Award, which he shared in 2007 with Richard Taylor.

DATE :

Thursday, February 11, 2016

TIME :

4:00 p.m.

PLACE :

Concordia University, Library Building, 9th floor, Salle/Room LB 921-04

1400 De Maisonneuve West

Friday, February 5, 2016

**Chain reactions**

To every action, there is an equal and opposite reaction. However, there turn out to exist in nature situations where the reaction seems to be neither equal in magnitude nor opposite in direction to the action. We will see a series of table-top demos and experimental movies, apparently in more and more violation of Newton's 3rd law, and give a full analysis of what is happening, discovering in the end that this phenomenon are in a sense generic. The keys are shock, singular material property, and supply of "critical geometry".

**Date / Time ** : Friday, February 5, 2016 - 16:00

** Venue ** : UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, room PK-5115

Friday, January 29, 2016

**Stability and instability for nonlinear elliptic PDE with slight variations to the data**

We will consider the question of stability of solutions to nonlinear elliptic PDE when slightly varying the data. We will take as a model the Standing Wave Equation for critical nonlinear Schrödinger and Klein-Gordon Equations on a closed manifold, and we will look at variations to the potential functions in these equations. A number of results have been obtained on this question in the last two decades, and we now have an accurate picture of the stability and instability of solutions to these equations. I will give an overview of these results and explain why certain types of unstable solutions can exist for some potential functions or in some geometries, and not others.

**Date / Time ** : Friday, January 29, 2016 - 16:00

** Venue ** : CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, room 6214

Friday, January 22, 2016

**Big data & mixed-integer (non linear) programming**

In this talk I review a couple of applications on Big Data that I personally like and I try to explain my point of view as a Mathematical Optimizer -- especially concerned with discrete (integer) decisions -- on the subject. I advocate a tight integration of Data Mining, Machine Learning and Mathematical Optimization (among others) to deal with the challenges of decision-making in Data Science. Those challenges are the core of the mission of the Canada Excellence Research Chair in "Data Science for Real-time Decision Making" that I hold.

**Date / time: **Friday, January 22, 2016 - 4:00 pm

**Venue:** UQAM, President-Kennedy Building, 201, ave du Président-Kennedy, room PK-5115

Friday, January 15, 2016

**Maximum of strongly correlated random variables **

One of the main goal of probability theory is to find "universal laws". This is well-illustrated by the Law of Large Numbers and the Central Limit Theorem, dating back to the 18th century, which show convergence of the sum of random variables with minimal assumptions on their distributions. Much of current research in probability is concerned with finding universal laws for the maximum of random variables. One universality class of interest (in mathematics and in physics) consists of stochastic processes whose correlations decay logarithmically with the distance. In this talk, we will survey recent results on the subject and their connection to problems in mathematics such as the maxima of the Riemann zeta function on the critical line and of the characteristic polynomial of random matrices.

** The talk will be given in french with English slides. **

Coffee will be served before the conference and a reception will follow at Salon Maurice-L’Abbé (Room 6245).

**Date and time:** Friday, January 15, 2016, 16:00 - 17:00

**Venue:** Room 6254, Centre de recherches mathématiques, Pavillon André-Aisenstadt, 2920, chemin de la Tour

Friday, January 8, 2016

The Seminars in Undergraduate Mathematics in Montreal (SUMM) is an annual event organized by students who are currently enrolled in an undergraduate mathematics program at one of the four Montreal universities.

The 2016 edition of SUMM will be held at Université du Québec à Montréal (UQÀM) on January 8-9-10.

**Keynote Speakers :**

– Dimiter Dryanov, Department of Mathematics and Statistics, Concordia University.

– Marlène Frigon, Département de Mathématiques et de Statistique, Université de Montréal.

– Christian Genest, Department of Mathematics and Statistics, McGill University.

– Franco Saliola, Département de Mathématiques, Université du Québec à Montréal.

For more information, please contact us.

Thursday, December 10, 2015

**Causal discovery with confidence using invariance principles**

What is interesting about causal inference? One of the most compelling aspects is that any prediction under a causal model is valid in environments that are possibly very different to the environment used for inference. For example, variables can be actively changed and predictions will still be valid and useful. This invariance is very useful but still leaves open the difficult question of inference. We propose to turn this invariance principle around and exploit the invariance for inference. If we observe a system in different environments (or under different but possibly not well specified interventions) we can identify all models that are invariant. We know that any causal model has to be in this subset of invariant models. This allows causal inference with valid confidence intervals. We propose different estimators, depending on the nature of the interventions and depending on whether hidden variables and feedbacks are present. Some empirical examples demonstrate the power and possible pitfalls of this approach.

**Date / Time ** : Thursday, December 10,2015 - 3:30 PM

** Venue ** : UdeM, Pav. Roger-Gaudry, salle S-116

Friday, December 4, 2015

The Canadian Mathematical Society (CMS) invites the mathematical community to the 2015 CMS Winter Meeting in Montreal, Quebec, from December 4-7. All meeting activities are taking place at the Hyatt Regency Montreal (1255 Jeanne-Mance, Montreal, Quebec, Canada, H5B 1E5).

Friday, November 27, 2015

**Measuring irregularities in data : Can fractals help to classify Van Gogh paintings?**

Benoît Mandelbrot defined fractal geometry as the geometry of irregular sets; he and his followers successfully used the mathematical concepts of fractional dimensions to quantify this irregularity and thus popularized new classification tools among scientists working in many disciplines. Recently, these ideas have proved very fruitful in multifractal analysis, which deals with the analysis of irregular functions. We will show how the seminal ideas introduced in fractal geometry have been diverted in order to supply new classification tools for signals and images, and we will present a selected choice of applications including: - Model classification in the context of fully developed turbulence and the diagnostic of heart-beat failure. - Modeling of internet flowl - Stylometry tools helping art historians to differentiate between the paintings of several masters.

**Date / Time ** : Friday, November 27, 2015 - 16:00

** Venue ** : CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

Thursday, November 26, 2015

**Inference regarding within-family association in disease onset times under biased sampling schemes**

In preliminary studies of the genetic basis for chronic conditions, interest routinely lies in the within-family dependence in disease status. When probands are selected from disease registries and their respective families are recruited, a variety of ascertainment bias-corrected methods of inference are available which are typically based on models for correlated binary data. This approach ignores the age that family members are at the time of assessment. We consider copula-based models for assessing the within-family dependence in the disease onset time and disease progression, based on right-censored and current status observation of the non-probands. Inferences based on likelihood, composite likelihood and estimating functions are each discussed and compared in terms of asymptotic and empirical relative efficiency. This is joint work with Yujie Zhong.

**Date / Time ** : Thursday, November 26, 2015 - 3:30 PM

** Venue ** : McGill, Burnside Hall, room 306

Friday, November 20, 2015

**Sur l'étude des singularités dans des modèles mathématiques de cristaux liquides**

L'analyse de modèles mathématiques pour les cristaux liquides pose beaucoup de défis, vu leurs proximités à l'étude des singularités dans les applications harmoniques. Dans ce colloque, je vais présenter des modèles mathématiques utilisés dans l'étude des cristaux liquides, la connexion avec les résultats classiques pour les applications harmoniques, ainsi que les nouvelles méthodes utilisées pour étudier les singularités dans le modèle de Landau-de Gennes. Ce modèle permet une plus grande variété de singularités que le modèle d'Oseen-Frank basé sur les applications harmoniques à valeur dans la sphère. (The talk will delivered in French with English slides.)

**Date / Time ** : Friday, November 20, 2015 - 4:00 PM

** Venue ** : UQAM, Sherbrooke Building, Room SH-2420

Friday, November 13, 2015

**Random walks in random environments**

The goal of this talk is to present some recent developments in the field of random walks in random environments. We chose to do this by presenting a specific model, known as biased random walk on Galton-Watson trees, which is intuitively easy to understand but gives rise to many interesting and challenging questions. We will then explain why this model is actually representative of a whole class of models which exhibit universal limiting behaviours.

**Date / Time ** : Friday, November 13, 2015 - 16:00

** Venue ** : CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

Friday, November 6, 2015

**Walls in random groups**

I will give an overview of Gromov's density model for random groups. These groups are hyperbolic and for large densities are exotic enough to have Kazhdan's property (T). I will focus on small densities and explain the techniques of Ollivier and Wise, and Mackay and myself to tame these groups by finding "walls" and hence an action on a CAT(0) cube complex.

**Date:** Friday, November 6, 2015

**Time:** 4:00 PM

**Venue:** UQAM, Pavillon Sherbrooke, Room SH-2420

*The talk will be followed by a wine and cheese reception.*

Friday, October 30, 2015

**A knockoff filter for controlling the false discovery rate**

The big data era has created a new scientific paradigm: collect data first, ask questions later. Imagine that we observe a response variable together with a large number of potential explanatory variables, and would like to be able to discover which variables are truly associated with the response. At the same time, we need to know that the false discovery rate (FDR)---the expected fraction of false discoveries among all discoveries---is not too high, in order to assure the scientist that most of the discoveries are indeed true and replicable. We introduce the knockoff filter, a new variable selection procedure controlling the FDR in the statistical linear model whenever there are at least as many observations as variables. This method works by constructing fake variables, knockoffs, which can then be used as controls for the true variables; the method achieves exact FDR control in finite sample settings no matter the design or covariates, the number of variables in the model, and the amplitudes of the unknown regression coefficients, and does not require any knowledge of the noise level. This is joint work with Rina Foygel Barber.

**Date / Time ** : Friday, October 30, 2015 - 16:00

** Venue ** : CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 1360

Friday, October 23, 2015

**Weighted Hurwitz Numbers: Classical and Quantum**

The study of Hurwitz numbers, which enumerate branched coverings of the Riemann sphere, is classical, going back to the pioneering work of Hurwitz in the 1880’s. There is an equivalent combinatorial problem, related by monodromy that was developed by Frobenius in his pioneering work on character theory, consisting of enumeration of factorizations of elements of the symmetric group. In 2000, Okounkov and Pandharipande began their program relating Hurwitz numbers to other combinatorial/topological invariants associated to Riemann surfaces, such as as Gromov-Witten and Donaldson-Thomas invariants. This has since been further developed by others to include, e.g., Hodge invariants and relations to knot invariants. A key result of Okounkov and Pandharipande was to express the generating functions for special classes of Hurwitz numbers, e.g., including only simple branching, plus one, or two other branch points, as special types of Tau functions of integrable hierarchies such as Sato's KP hierarchy and Takasaki-Takebe’s 2D Toda lattice hierarchy, together with associated semi-infinite wedge product representations. The differential/algebraic equations satisfied by such generating functions provide a new perspective, implying deep interrelations between these various types of enumerative invariants. In more recent work, these ideas have been extended to include generating functions for a very wide class of branched coverings, with suitable combinatorial interpretations, including broad class of weighted enumerations that select amongst infinite parametric families of weights. These make use not only of the six standard bases for the ring of symmetric functions, such as Schur functions, and monomomial sum symmetric functions, but also their “quantum” deformations, involving the pair of deformation parameters (q,t) appearing the in theory of Macdonald polynomials. The general theory of weighted Hurwitz numbers, together with various applications and examples coming from Random Matrix theory and enumerative geometry will be explained in a simple, unified way, based on special elements of and bases for the center of the symmetric group algebra, and the characteristic map to the ring of symmetric polynomials. The simplest quantum case provides a relation between special weighted enumerations of branched coverings and the statistical nechanics of Bose-Eintein gases. Various other specializations, to such bases as: Hall-Littlewood, Jack, q-Whittaker, dual q-Whttaker as well as certain special classical weightings have further applications, in physics, geometry, group theory and combinatorics.

**Date / Time ** : Friday, October 23, 2015 - 16:00

** Venue ** : UQAM - Sherbrooke Building, Room SH-2420 (one floor below the normal colloquium room)

**A wine and cheese reception will follow the talk. **

Friday, October 16, 2015

**Holomorphic functions, convexity and transversality**

Morse theory is a powerful tool to study the topology of real manifolds. After recalling its basic features, we will discuss the existence, on complex manifolds, of holomorphic functions giving similar information on the topology. More specifically, we will review the notions of pseudoconvexity and of Stein manifold so as to gradually explain the significance of a recent result, jointly obtained with John Pardon, which shows that any Stein domain can be presented as a Lefschetz fibration over the disk. The talk will be aimed at a general mathematical audience.

**Date / Time ** : Friday, October 16, 2015 - 16:00

** Venue ** : CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6254

Friday, October 9, 2015

**Coxeter Groups and Quiver Representations**

It has been understood since almost the beginning of the development of quiver representations, in the 1970s, that there are important connections between Coxeter groups and quiver representations. Nonetheless, further relations continue to appear. I will touch on the classical connections and some of the more recent ones, including the example of the parallel elaboration of the closely related concepts of exceptional sequences of representations and factorizations of Coxeter elements.

**Date / Time ** : Friday, October 9, 2015 - 16:00

** Venue ** : UQAM - Sherbrooke Building, Room SH-2420 (one floor below the normal colloquium room)

**A wine and cheese reception will follow the talk. **

Friday, September 25, 2015

**Analysis of first order systems of PDEs on manifolds without boundary**

In layman's terms a typical problem in this subject area is formulated as follows. Suppose that our universe has finite size but does not have a boundary. An example of such a situation would be a universe in the shape of a 3-dimensional sphere embedded in 4-dimensional Euclidean space. And imagine now that there is only one particle living in this universe, say, a massless neutrino. Then one can address a number of mathematical questions. How does the neutrino field (solution of the massless Dirac equation) propagate as a function of time? What are the eigenvalues (stationary energy levels) of the particle? Are there nontrivial (i.e. without obvious symmetries) special cases when the eigenvalues can be evaluated explicitly? What is the difference between the neutrino (positive energy) and the antineutrino (negative energy)? What is the nature of spin? Why do neutrinos propagate with the speed of light? Why are neutrinos and photons (solutions of the Maxwell system) so different and, yet, so similar? The speaker will approach the study of first order systems of PDEs from the perspective of a spectral theorist using techniques of microlocal analysis and without involving geometry or physics. However, a fascinating feature of the subject is that this purely analytic approach inevitably leads to differential geometric constructions with a strong theoretical physics flavour. References [1] See items 98-101, 103 and 104 on my publications page http://www.homepages.ucl.ac.uk/~ucahdva/publicat/publicat.html [2] Futurama TV series, Mars University episode (1999): Fry: Hey, professor. What are you teaching this semester? Professor Hubert Farnsworth: Same thing I teach every semester. The Mathematics of Quantum Neutrino Fields. I made up the title so that no student would dare take it.

**Date / Time ** : Friday, September 25, 2015 - 16:00

** Venue ** : UQAM - Sherbrooke Building, Room SH-2420 (one floor below the normal colloquium room)

**A wine and cheese reception will follow the talk. **

Monday, June 15, 2015

McGill University will host the CRM-PIMS probability summer school from June 15-July 11, 2015.

There will be two main courses, given by Alice Guionnet and Remco van der Hofstad, as well as mini-courses by Louigi Addario-Berry, Shankar Bhamidi and Jonathan Mattingly.

For more details, see: http://problab.ca/ssprob2015/index.php

Monday, June 15, 2015

The 2015 Séminaire de Mathématiques Supérieures will feature about a dozen minicourses on geometry of eigenvalues, geometry of eigenfunctions, spectral theory on manifolds with singularities, and computational spectral theory. There has been a number of remarkable recent developments in these closely related fields. The goal of the summer school is to shed light on different facets of modern spectral theory and to provide a unique opportunity for graduate students and young researchers to get a "big picture" of this rapidly evolving area of mathematics. The lectures will be given by the leading experts in the subject. The minicourses will be complemented by guided exercises sessions, as well as by several invited talks by the junior participants who have already made important contributions to the field. A particularly novel aspect of the school is the emphasis on the interactions between spectral geometry and computational spectral theory. We do not assume that the students are familiar with computational methods, and therefore we intend to provide tutorials where the participants will learn to develop and implement algorithms for numerical analysis of eigenvalue problems.

Friday, May 15, 2015

The 18th edition of the ISM Student Conference will be held at HEC Montreal from May 15 to 17, 2015. You can present your work (deadline for abstract submission: April 15, 2015) or simply attend the presentations and enjoy the networking activities. The keynote speakers are: Nantel Bergeron (York University), Matt Davison (University of Western Ontario), Stephen Fienberg (Carnegie Mellon), and the 2015 Carl Herz Prize winner. For more information or to register, please visit the conference web site: http://www.crm.umontreal.ca/2015/ISM2015/index_e.php.

Friday, May 8, 2015

This year, we are celebrating the international year of light by welcoming John Dudley, instigator of the international year. You are invited to a half-day of activties where you will discover the role of light in our civilization and how mathematics allows us to study it. In French. Free registration. Further information.

Thursday, April 9, 2015

**Modular generating series and arithmetic geometry**

I will survey the development of the theory of theta series and describe some recent advances/work in progress on arithmetic theta series. The construction and modularity of theta series as counting functions for lattice points for positive definite quadratic forms is a beautiful piece of classical mathematics with its origins in the mid 19th century. Siegel initiated the study of the analogue for indefinite quadratic forms. Millson and I introduced a geometric variant in which the theta series give rise to modular generating series for the cohomology classes of "special" algebraic cycles on locally symmetric varieties. These results motivate the definition of analogous generating series for the classes of such special cycles in the Chow groups and for the classes in the arithmetic Chow groups of their integral extensions. The modularity of such series is a difficult problem. I will discuss various cases in which recent progress has been made and some of the difficulties involved.

**Date / Time ** : Thursday, April 9, 2015 - 4:00 PM

** Venue ** : CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, room 6254

Thursday, April 2, 2015

**A combinatorial approach to dynamics applied to switching networks**

Models of multiscale systems, such as those encountered in systems biology, are often characterized by heuristic nonlinearities and poorly defined parameters. Furthermore, it is typically not possible to obtain precise experimental data for these systems. Nevertheless, verification of the models requires the ability to obtain meaningful dynamical structures that can be compared quantitatively with the experimental data. With this in mind we present a purely combinatorial approach to modeling dynamics. We will discuss this approach in the context of switching networks.

**Date / Time**: Thursday, April 2, 2015 - 4:30 PM

** Venue**: Université de Sherbrooke

Thursday, April 2, 2015

**Uniqueness of blowups and Lojasiewicz inequalities**

The mean curvature flow (MCF) of any closed hypersurface becomes singular in finite time. Once one knows that singularities occur, one naturally wonders what the singularities are like. For minimal varieties the first answer, by Federer-Fleming in 1959, is that they weakly resemble cones. For MCF, by the combined work of Huisken, Ilmanen, and White, singularities weakly resemble shrinkers. Unfortunately, the simple proofs leave open the possibility that a minimal variety or a MCF looked at under a microscope will resemble one blowup, but under higher magnification, it might (as far as anyone knows) resemble a completely different blowup. Whether this ever happens is perhaps the most fundamental question about singularities. We will discuss the proof of this long standing open question for MCF at all generic singularities and for mean convex MCF at all singularities. This is joint work with Toby Colding.

**Date / Time ** :Thursday, April 2, 2015 - 4:00 PM

** Lieu ** : McGill University, Burnside Hall, 805 rue Sherbrooke 0., Montréal, room 920

Thursday, March 26, 2015

**Left-orderings of groups and the topology of 3-manifolds**

Many decades of work culminating in Perelman's proof of Thurston's geometrisation conjecture showed that a closed, connected, orientable, prime 3-dimensional manifold *W* is essentially determined by its fundamental group π_{1}(*W*). This group consists of classes of based loops in *W* and its multiplication corresponds to their concatenation. An important problem is to describe the topological and geometric properties of *W* in terms of π_{1}(*W*). For instance, geometrisation implies that *W* admits a hyperbolic structure if and only if π_{1}(*W*) is infinite, freely indecomposable, and contains no **Z** ⊕ **Z** subgroups. In this talk I will describe recent work which has determined a surprisingly strong correlation between the existence of a left-order on π_{1}(W) (a total order invariant under left multiplication) and the following two measures of largeness for *W*:

a) the existence of a co-oriented taut foliation on *W* - a special type of partition of *W* into surfaces which fit together locally like a deck of cards.

b) the condition that *W* not be an L-space - an analytically defined condition representing the non-triviality of its Heegaard-Floer homology.

I will introduce each of these notions, describe the results which connect them, and state a number of open problems and conjectures concerning their precise relationship.

**Date / Time**: Thursday, March 26, 2015 - 4:00 PM

** Venue**: McGill University, Burnside Hall, 805 rue Sherbrooke 0., Montréal, room 920

Thursday, March 19, 2015

**Integrable probability**

The goal of the talk is to survey the emerging field of integrable probability, whose goal is to identify and analyze exactly solvable probabilistic models. The models and results are often easy to describe, yet difficult to find, and they carry essential information about broad universality classes of stochastic processes.

**Date / Time**: Thursday, March 19, 2015 - 4:00 PM

** Venue**: McGill University, Burnside Hall, 805 rue Sherbrooke 0., Montréal, room 920

Thursday, March 12, 2015

**The upper half-planes**

The upper half-planes (complex and p-adic) are very elementary objects, but they have a surprisingly rich structure that I will explore in the talk.

**Date / Time**: Thursday, March 12, 2015 - 4:00 PM

** Venue**: CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, room 1360

Thursday, March 5, 2015

**Periods**

We will discuss periods, in particular the periods conjecture of Kontsevich and Zagier and the relationship between formal periods and Nori motives.

**Date / Time **: Thursday, March 5, 2015 - 4:00 PM

** Venue ** : McGill University, Burnside Hall, 805 rue Sherbrooke 0., Montréal, room 920

Thursday, February 26, 2015

**Categorification in representation theory**

This will be an expository talk concerning the idea of categorification and its role in representation theory. We will begin with some very simple yet beautiful observations about how various ideas from basic algebra (monoids, groups, rings, representations etc.) can be reformulated in the language of category theory. We will then explain how this viewpoint leads to new ideas such as the "categorification" of the above-mentioned algebraic objects. We will conclude with a brief synopsis of some current active areas of research involving the categorification of quantum groups. One of the goals of this idea is to produce four-dimensional topological quantum field theories. Very little background knowledge will be assumed.

**Date / Time**: Thursday, February 26, 2015 - 4:00 PM

** Venue**: McGill University, Burnside Hall, 805 rue Sherbrooke 0., Montréal, room 920

Thursday, February 19, 2015

**Irrationality proofs, moduli spaces and dinner parties**

After introducing an elementary criterion for a real number to be irrational, I will discuss Apery’s famous result proving the irrationality of zeta(3). Then I will give an overview of subsequent results in this field, and finally propose a simple geometric interpretation based on a classical dinner party game.

**Date / Time ** : Thursday, February 19, 2015 - 4:00 PM

** Venue ** : CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, room 6214

Thursday, February 12, 2015

**The role of boundary layers in the global ocean circulation**

Comprendre les mécanismes qui régissent la circulation océanique est un défi pour les géophysiciens, mais aussi pour les mathématiciens qui doivent développer de nouveaux outils d'analyse pour ces modèles complexes (qui font intervenir en particulier de très nombreuses échelles de temps et d'espace). Un mécanisme particulièrement important pour la circulation à l'échelle planétaire est le phénomène de couche limite qui explique une partie des échanges énergétiques. On montrera ici au travers d'un modèle très simplifié qu'il permet d'expliquer notamment l'intensification des courants de bord Ouest. On évoquera ensuite les difficultés mathématiques liées à la prise en compte de la géométrie. Note : l'exposé sera en anglais avec des transparents en français.

**Date / Time**: Thursday, February 12, 2015 - 4:00 PM

** Venue**: McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, room 920

Thursday, February 5, 2015

**Cobordism and Lagrangian topology**

This talk aims to discuss how two different basic organizing principles in topology come together in the study of Lagrangian submanifolds. The first principle is cobordism and it emerged in topology in the 1950’s, mainly starting with the work of Thom. It was introduced in Lagrangian topology by Arnold in the 1970’s. The second principle is to reconstruct a subspace of a given space from a family of "slices", each one obtained by intersecting the subspace with a member of a preferred class of special "test" subspaces. For instance, a subspace of 3d euclidean space can be described as the union of all its intersections with horizontal planes. The key issue from this point of view is, of course, how to assemble all the slices together. The perspective that is central for my talk originates in the work of Gromov and Floer in the 1980’s: if the ambient space is a symplectic manifold M, and if the subspace to be described is a Lagrangian submanifold, then, surprisingly,the "glue" that puts the slices together in an efficient algebraic fashion is a reflection of the combinatorial properties of J-holomorphic curves in M. This point of view has been pursued actively since then by many researchers such as Hofer, Fukaya, Seidel leading to a structure called the Fukaya category. Through recent work of Paul Biran and myself, cobordism and the Fukaya category turn out to be intimately related and at the end of the talk I intend to give an idea about this relation.

**Date / Time **: Thursday, February 5, 2015 - 4:00 PM**Venue **: McGill University, Burnside Hall, 805 rue Sherbrooke 0., Montréal, room 920

Thursday, January 29, 2015

** Spectres et pseudospectres**

Les valeurs propres sont parmi les notions les plus utiles en mathématiques: elles permettent la diagonalisation des matrices, elles décrivent l'asymptotique et la stabilité, elles donnent de la personnalité à une matrice. Cependant, lorsque la matrice en question n'est pas normale, l'analyse par des valeurs propres ne donne qu'une information très partielle, et peut même nous induire en erreur. Cet exposé se veut une introduction à la théorie des pseudospectres, un raffinement de la théorie spectrale standard qui s'est avéré utile dans des applications concernant des matrices non normales. Je vais m'intéresser surtout à la question suivante: À quel point les pseudospectres d'une matrice déterminent-ils le comportement de la matrice?

**Date / Time**: Thursday, January 29, 2015 - 4:00 PM

** Venue**: McGill University, Burnside Hall, 805 Sherbrooke Street West, Montréal, room 920

Thursday, January 22, 2015

**On the usefulness of mathematics for insurance risk theory - and vice versa**

This talk is on applications of various branches of mathematics in the field of risk theory, a branch of actuarial mathematics dealing with the analysis of the surplus process of a portfolio of insurance contracts over time. At the same time such practical problems frequently trigger mathematical research questions, in some cases leading to remarkable identities and connections. Next to the close interactions with probability and statistics, examples will include the branches of real and complex analysis, algebra, symbolic computation, number theory and discrete mathematics.

**Date / Time**: Thursday, January 22, 2015 - 4:00 PM

** Venue**: McGill University, Burnside Hall, 805 Sherbrooke Street West, Montréal, room 920

Thursday, January 15, 2015

**Functional data analysis and related topics**

Functional data analysis (FDA) has received substantial attention, with applications arising from various disciplines, such as engineering, public health, finance, etc. In general, the FDA approaches focus on nonparametric underlying models that assume the data are observed from realizations of stochastic processes satisfying some regularity conditions, e.g., smoothness constraints. The estimation and inference procedures usually do not depend on merely a finite number of parameters, which contrasts with parametric models, and exploit techniques, such as smoothing methods and dimension reduction, that allow data to speak for themselves. In this talk, I will give an overview of FDA methods and related topics developed in recent years.

**Date / Time ** : Thursday, January 15, 2015 - 4:00 PM

** Venue ** : CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, room 1360

Thursday, December 4, 2014

**Algebraic combinatorics and finite reflection groups**

The lecture will be delivered in French, with English slides, so that anyone may enjoy it. ----- La conférence sera présentée en français, avec des transparents en anglais, pour que tous puissent suivre. Les dernières années ont vu une explosion d’activités à la frontière entre la combinatoire algébrique, la théorie de la représentation et la géométrie algébrique, avec des liens captivants avec la théorie des nœuds et la physique mathématique. En gardant un large auditoire en tête, nous esquisserons en quoi cette interaction a été très fructueuse et a soulevé de nouvelles questions intrigantes dans les divers domaines concernés. Nous essaierons de donner la saveur des résultats obtenus, des techniques utilisées, du grand nombre de questions ouvertes, et du pourquoi de leur intérêt. Ce fascinant échange entre combinatoire et algèbre fait d’une part intervenir des généralisations au contexte des rectangles des « chemins de Dyck ». Il est bien connu, depuis Euler, que ces chemins sont comptés par les nombres de Catalan, dans le cas d’un carré. De plus, les fonctions de stationnement (parking functions) sont intimement reliées à ces chemins. D’autre part, du côté algébrique, apparaissent des S_{n}-module bigradué de polynômes harmoniques diagonaux du groupe symétrique S_{n}. Il a été conjecturé qu’une énumération adéquate des fonctions de stationnement, associées à certaines familles de chemins de Dyck, fournit une formule combinatoire explicite du caractère bigradué de ces modules. Cette conjecture, connue sous le nom de conjecture « shuffle », a récemment été grandement étendue pour couvrir tous les cas rectangulaires. Interviennent dans tout ceci, des opérateurs sur les polynômes de Macdonald, l’algèbre de Hall elliptique, les algèbres de Hecke affines doubles (DAHA), le schéma de Hilbert de points dans le plan, etc.

**Date / Time**: Thursday, December 4, 2014 - 4:00 PM

** Venue**: CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, room 6214

Thursday, November 27, 2014

**On the well-posedness of the 2D stochastic Allen-Cahn equation**

Non-linear parabolic PDE arise in many physical and biological settings; we often need to incorporate the effects of additive white noise. The resultant stochastic partial differential equations are well-understood in 1D. In higher spatial dimensions, there is an interesting dichotomy: such models are popular in application, while mathematicians assume these models to be ill-posed. We investigate the specific case of the two dimensional Allen-Cahn equation driven by additive white noise. Without noise, the Allen-Cahn equation is 'pattern-forming'. Does the presence of noise affect this behaviour? The precise notion of a weak solution to this equation is unclear. Instead, we regularize the noise and introduce a family of approximations. We discuss the continuum limit of these approximations and show that it exhibits divergent behavior. Our results show that a series of published numerical studies are somewhat problematic: shrinking the mesh size in these simulations does not lead to the recovery of a physically meaningful limit. This is joint work with Marc Ryser and Paul Tupper.

**Date / Time ** : Thursday, November 27, 2014 - 3:30 PM

** Venue ** : Laval University, Alexandre Vachon Building, room 2830

Thursday, November 20, 2014

**High-dimensional phenomena in mathematical statistics and convex analysis**

Statistical models in which the ambient dimension is of the same order or larger than the sample size arise frequently in different areas of science and engineering. Although high-dimensional models of this type date back to the work of Kolmogorov, they have been the subject of intensive study over the past decade, and have interesting connections to many branches of mathematics (including concentration of measure, random matrix theory, convex geometry, and information theory). In this talk, we provide a broad overview of the general area, including vignettes on phase transitions in high-dimensional graph recovery, and randomized approximations of convex programs.

**Date / Time ** : Thursday, November 20, 2014 - 4:00 PM

**Venue ** : CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, room 6214

Thursday, November 13, 2014

**Recent advances in the arithmetic of elliptic curves**

In the past few years there have been several spectacular advances in understanding the arithmetic of elliptic curves including results about ranks on average and on the conjecture of Birch and Swinnerton-Dyer. I will give an introduction to the main problems of interest and survey some of these developments. This talk will be addressed to a general mathematical audience.

**Date / Time ** : Thursday, November 13, 2014 - 4:00 PM

**Venue ** : CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, room 6214

Thursday, November 6, 2014

**The cubical route to understanding groups**

Cube complexes have come to play an increasingly central role within geometric group theory, as their connection to right-angled Artin groups provides a powerful combinatorial bridge between geometry and algebra. This talk will primarily aim to introduce nonpositively curved cube complexes, and then describe some of the developments that have recently culminated in the resolution of the virtual Haken conjecture for 3-manifolds, and simultaneously dramatically extended our understanding of many infinite groups.

**Date / Time **: Thursday, November 6, 2014 - 4:00 PM**Venue **: CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, room 6214

Thursday, October 30, 2014

**A pedestrian approach to group representations**

Determining the number of walks of *n* steps from vertex A to vertex B on a graph often involves clever combinatorics or tedious treading. But if the graph is the representation graph of a group, representation theory can facilitate the counting and provide much insight. This talk will focus on connections between Schur-Weyl duality and walking on representation graphs. Examples of special interest are the simply-laced affine Dynkin diagrams, which are the representation graphs of the finite subgroups of the special unitary group SU(2) by the McKay correspondence. The duality between the SU(2) subgroups and certain algebras enables us to count walks and solve other combinatorial problems, and to obtain connections with the Temperley-Lieb algebras of statistical mechanics, with partitions, with Stirling numbers, and much more.

**Date / Time ** : Thursday,October 30, 2014 - 4:00 PM

**Venue**: CRM, Université de Montréal, Pav. André-Aisenstadt, 2920, ch. de la Tour, rom 6214

Thursday, October 9, 2014

**Applications of additive combinatorics to homogeneous dynamics**

We will discuss the role played by additive combinatorics in attacks on various problems in dynamics related to finer equidistribution questions beyond Duke's Theorem, particularly those posed by McMullen and Einsiedler-Lindenstrauss-Michel-Venkatesh. This work is joint with Jean Bourgain.

**Date / Time**: Thursday, October 9, 2014 - 4:00 PM

** Venue**: CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, room 1140

Thursday, October 2, 2014

**Universality in random matrix theory**

Wigner stated the general hypothesis that the distribution of eigenvalue spacings of large complicated quantum systems is universal, in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. The simplest case for this hypothesis concerns large but finite dimensional matrices. I will explain some historical aspects random matrix theory, as well as recent techniques developed to prove eigenvalues and eigenvectors universality, for matrices with independent entries from all symmetry classes. The methods are both probabilist (random walks and coupling) and analytic (homogenization for parabolic PDEs).

**Date / Time ** : Thursday, October 2, 2014 - 16:00

** Venue ** : CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

Friday, August 29, 2014

A conference in honor of Louis-Paul Rivest will be held at Université Laval on August 28 and 29 to mark his 60th birthday and his many contributions to science.

We hope that you can join us. Please register at http://www.crm.umontreal.ca/2014/Rivest14/index.php

Monday, June 23, 2014

Counting objects of arithmetic interest (such as quadratic forms, number fields, elliptic curves, curves of a given genus, ...) in order of increasing arithmetic complexity, is among the most fundamental enterprises in number theory, going back (at least) to the fundamental work of Gauss on composition of binary quadratic forms and class groups of quadratic fields.

In the past decade tremendous progress has been achieved, notably through Bhargava's revolutionary program blending elegant algebraic techniques with powerful analytic ideas. It suffices to mention the striking upper bounds on the size of Selmer groups (and therefore ranks) of elliptic curves and even Jacobians of hyperelliptic curves of higher genus, among the many other breakthroughs that have grown out of this remarkable circle of ideas.

The 2014 Summer School will be devoted to covering these recent developments, with the objective of attracting researchers who are in the early stages of their career into this active and rapidly developing part of number theory.

For more information, view the website.

Friday, May 16, 2014

This annual conference is an occasion Québec mathematics and statistics students to meet for one weekend. This year it will be held May 16-18, 2014. All are invited to present their current research or another subject matter judged worthy of interest.

Twenty-minute student presentations are an excellent way to discover a diversity of subjects and exchange ideas with fellow students. We strongly encourage participants to present in French, however, presentations in English are always welcome. In addition to the student conferences, there will be 50-minute plenary lectures by well-known professors.

Friday sessions will finish with a wine and cheese reception so that we can all meet and spend a very pleasant weekend.

The 2014 colloque panquébécois des étudiants de l'ISM will be held at Université Laval in Québec City.

We hope to see you this spring!