The analysis group is affiliated with the CRM Mathematical Analysis Laboratory which organizes many scientific events. Current research interests of the members of this group may be roughly classified under the following headings:

- Analysis on Manifolds: spectral geometry (eigenvalues and eigenfunctions of Laplacians), quantum chaos.
- Classical Analysis
- Complex Analysis: complex approximation, discrete two-generator groups, complex dynamics, several complex variables, analytic multifunctions.
- Ergodic Theory: spectral theory of measure preserving transformations, Baire category results in ergodic theory, generalizations of the pointwise ergodic theorems to sequences of generalized projections.
- Functional Analysis: Banach algebras, resolvents and controllability of operators, generalized spectral theorem and sequences of self-adjoint operators and their weak limits, matrix analysis and inequalities, spectral theory and mathematical physics.
- Harmonic Analysis: trigonometric series, automorphic forms, singular integrals, Fourier transforms, multiplier operators, Littlewood-Paley theory, harmonic functions on Rn, Hardy spaces, square functions, connections to probability theory and to ergodic theory.
- Partial Differential Equations: connections to functional, geometric and harmonic analysis.
- Potential Theory: duality in potential theory, harmonic approximation, boundary behaviour, potential theory on trees.

- Line Baribeau (Laval)
- Galia Dafni (Concordia)
- S.W. Drury (McGill)
- Richard Fournier (CRM, Dawson College)
- Marlène Frigon (UdeM)
- Paul M. Gauthier (UdeM)
- Alexandre Girouard (Laval)
- Frédéric Gourdeau (Laval)
- Pengfei Guan (McGill)
- Dmitry Jakobson (McGill)
- Vojkan Jaksic (McGill)
- Damir Kinzebulatov (Laval)
- Alexey Kokotov (Concordia)
- Paul Koosis (McGill)
- Javad Mashreghi (Laval)
- Iosif Polterovich (UdeM)
- Thomas Ransford (Laval)
- Dominic Rochon (UQTR)
- Jérémie Rostand (Laval)
- Alexander Shnirelman (Concordia)
- Alina Stancu (Concordia)
- J.C. Taylor (McGill)
- John Toth (McGill)
- Jérôme Vétois (McGill)

This program is designed to introduce students to research in the broad area of analysis, ranging from classical analysis to modern analysis, with applications in such fields as geometry, mathematical physics, number theory, and statistics.

It is very important for students interested in the analysis program to follow one of the following sequences of introductory graduate level analysis courses. These courses provide the necessary preparation for the more advanced courses offered by the program.

Measure Theory (Concordia MAST 669)

Functional Analysis I (Concordia MAST 662)

or

Advanced Real Analysis I (McGill MATH-564)

Advanced Real Analysis II (McGill MATH-565)

Advanced Complex Analysis (McGill MATH-566)

or

Mesure et intégration (Université de Montréal MAT 6111)

Analyse fonctionnelle (Université de Montréal MAT 6112)

Topologie générale (Université de Montréal MAT 6310)

Analyse complexe: sujets spéciaux (Université de Montréal MAT 6182K)

or

Analyse fonctionnelle I (Laval MAT-7100)

Théorie de la mesure et intégration (Laval MAT-6000)

Équations aux derivées partielles (Laval MAT-7220)

Starting with classical inequalities for convex sets and functions, the course aims to present famous geometric inequalities like the Brunn-Minkowski inequality and its related functional form, Prekopa-Leindler, the Blaschke-Santalo inequality, the Urysohn inequality, as well as more modern ones such as the reverse isoperimetric inequality, or the Brascamp-Lieb inequality and its reverse form. In the process, we will touch upon log-convex functions, duality for sets and functions and, generally, extremum problems.

Review of theory of measure and integration; product measures, Fubini's theorem; Lp spaces; basic principles of Banach spaces; Riesz representation theorem for C(X); Hilbert spaces; part of the material of MATH 565 may be covered as well.

The course is devoted to the basics of the theory of operators in Hilbert space with emphasis on applications to Partial Differential Equations.

**Prerequisites**: MATH 564, MATH 565, and MATH 566.

**Official Book:** Haim Brezis, Functional Analysis, Sobolev spaces and partial differential equations. Since it is available for download, this book has not been ordered from the bookstore.

Please note that there are many sets of notes for functional analysis on the web. See the course webpage for a list and links.

**Official Syllabus:** Banach spaces. Hilbert spaces and linear operators on these. Spectral theory. Banach algebras. A brief introduction to locally convex spaces.

**Proposed Syllabus:** Banach and Hilbert spaces, theorems of Hahn-Banach and Banach-Steinhaus, open mapping theorem, closed graph theorem, Fredholm theory, spectral theorem for compact self- adjoint operators, spectral theorem for bounded self-adjoint operators, Banach algebras and the Gelfand theory, Locally convex spaces. Additional topics to be chosen from: Lorentz spaces and interpolation, distributions and Sobolev spaces, The von Neumann-Schatten classes, symbolic calculus of Hilbert space operators, representation theory and harmonic analysis, semigroups of operators, Krein-Milman theorem, tensor products of Hilbert spaces and Banach spaces, fixed point theorems.

**Assessment:** 40% assignments, 60% final exam.

**Exam:** The final examination will be a take-home exam. There is no "additional work" option and the grade of incomplete will not be given. A supplemental exam will be available.

**Course webpage:** http://www.math.mcgill.ca/drury/teaching/math635f18/math635f18.php

**Exam Viewing:** The instructor reserves the right to set a specific time or times for the purpose of exam viewing. If such times are set, they will be announced on the course webpage.

**Note:** In the event of extraordinary circumstances beyond the University's control, the content and/or evaluation scheme in this course is subject to change.

**Note:** In accord with McGill University's Charter of Students' Rights, students in this course have the right to submit in English or in French any written work that is to be graded.

**Academic Integrity:** McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/integrity for more information).

Contenu du cours: ensembles mesurables, mesure de Lebesgue; principes de Littlewood, théorèmes de Lusin et de Egorov; intégrale de Lebesgue, théorème de Fubini, espaces L_{1} et L_{2}; mesures absolument continues, théorème de Radon-Nikodym; éléments de la théorie ergodique; mesure et dimension de Hausdorff, ensembles fractales.

- Espaces métriques
- Topologiques, d'Hilbert, de Banach
- Théorèmes de Hahn-Banach, de Banach-Steinhaus et du graphe fermé
- Topologies faibles
- Espaces réflexifs
- Décomposition spectrale des opérateurs auto-adjoints compacts.

This graduate course is an introduction to the treatment of nonlinear differential equations, and more generally to the theory of dynamical systems. The objective is to introduce the student to the theory of dynamical systems and its applications. Firstly, classical dynamics analysis techniques will be presented: continuous and discrete flows, existence and stability of solutions, invariant manifolds, bifurcations and normal forms. Secondly, an introduction to ergodic theory and an overview of modern applications will be presented: chaotic dynamics, strange attractors, dynamic entropy, high-dimensional systems (e.g. networks), driven dynamics and information processing. Particular attention will be paid to computations performed by dynamical systems.

At the end of the course, the student will be able to apply dynamical systems analysis techniques to concrete problems, as well as navigate the modern dynamical systems literature. Several examples and applications making use of numerical simulations will be used. To take this course, the student must master, at an undergraduate level, notions of calculus, linear differential equations, linear algebra and probability.

**Course schedule :**

Tuesday : 10h30 - 12h00, 5183 Pav. Andre-Aisenstadt

Thursday : 10h30 - 12h00, 5183 Pav. Andre-Aisenstadt

Équation des ondes, problème de Sturm-Liouville, distributions et transformation de Fourier, équation de Laplace, espaces de Sobolev, valeurs et fonctions propres du laplacien, éléments de la théorie spectrale, équation de la chaleur.

The Non-Standard Analysis (NSA) is a new mathematical discipline created by Abraham Robinson in 1960-s. The mathematical universe of NSA is immensely wider than that of the classical mathematics. It contains infinite and infinitesimal numbers, and much more. The NSA provides the tools of mathematical constructions which are infinitely more powerful than the classical ones. However, the use of NSA requires a strong logical discipline; not all the classical constructions are permitted there. We cannot form viable sets as freely as we are used to.

The course is devoted to the systematic study of the basics of NSA from the beginning, i.e. the set theory and elements of mathematical logic. On this basis the fundamental notions of NSA are developed, like the Transfer Principle, Internal Sets, Overflow and Underflow, etc. By the end of the course I'm going to discuss some applications of NSA to the Geometry (Asymptotic spaces) and Dynamics, including some problems of Fluid Dynamics.

The course does not require special prerequisits beyond the elementary set theory and analysis.

This course will discuss recent developments in ergodic theory and infinite groups, with an emphasis on the orbit structure of group actions (a subject often referred to as orbit equivalence).

Topics will include:

• invariant means on L^infinity

• orbit equivalence

• von Neumann algebras

• the cost theorem

• entropy

• Kazhdan’s property T and rigidity theorems (of Zimmer, Furman, etc.) • percolation theory relevant to orbit equivalence

**References:** No textbook is required for this course. Some information can be found in the following references:

1. V.F.R. Jones. von Neumann algebras.

2. A. Kechris, B. Miller. Topics in orbit equivalence theory. 3. R.J. Zimmer. Ergodic theory and semisimple groups.

**Prerequisites:** Some topics from Algebra 3 (the basics of group theory) and Analysis 3 (namely, topological spaces and the basics of measure theory).

**Grading:** 40% Assignments + 60% Final report.

Ce cours est une introduction au flot de Ricci. Nous étudierons les propriétés de base du flot (formules de variations, existence et unicité en temps court) et des applications du flot en géometrie et topologie (uniformisation des surfaces, les variétés de dimension 3 à courbure de Ricci positive).