Analysis

Program Description

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.

Program Members

Academic program

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.

Prerequisites:

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)

2018-19 Course Listings

Fall

Convex and Nonlinear Analysis

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.

Prof. Alina Stancu

MAST 661 A / 837

Institution: Concordia University

Advanced Real Analysis 1

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.

Prof. John Toth

MATH 564

Institution: McGill University

Functional Analysis 1

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

Prof. Stephen Drury

MATH 635

Institution: McGill University

Mesure et intégration

Contenu du cours: ensembles mesurables,  mesure de Lebesgue; principes de Littlewood, théorèmes de Lusin et de Egorov; intégrale de Lebesgue, théorème de Fubini, espaces L1 et L2; mesures absolument continues, théorème de Radon-Nikodym; éléments de la théorie ergodique; mesure et dimension de Hausdorff, ensembles fractales.

Prof. Iosif Polterovich

MAT 6111

Institution: Université de Montréal

Analyse fonctionnelle I

  • 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.

Prof. Marlène Frigon

MAT 6112

Institution: Université de Montréal

Nonlinear Differential Equations

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

Prof. Guillaume Lajoie

MAT 6115

Institution: Université de Montréal

Winter

Équations aux dérivées partielles - Université de Montréal

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.

 

Prof. Egor Shelukhin

MAT 6110

Institution: Université de Montréal