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, 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!