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)

2025-26 Course Listings

Fall

Measure Theory

Topics include Lebesgue measure, measurable sets and functions; Lebesgue integral; Differentiation and integration; Lebesgue (Lp) spaces; Additional topics may be covered if time permits.

Prof. Maria Ntekoume

MAST 669/2 sec. D / 837D

Institution: Concordia University

Théorie de la mesure et intégration

Introduction : explication des raisons de l'introduction de l'intégrale de Lebesgue. Espaces mesurables. Intégrale : intégrale des fonctions simples, extension, théorème de convergence monotone, théorème de Fatou. Fonctions intégrales. Exemples classiques (Lebesgue, Lebesgue-Stieltjes, etc.). Théorème de la convergence dominée. Modes de convergence. Décompositions des mesures. Produits de mesures : théorèmes de Tonelli et Fubini. Théorème de Riesz et de Radon-Nicodym.

Prof. Jérémie Rostand

MAT 6005

Institution: Université Laval

Équations aux dérivées partielles - Université Laval

Ce cours porte sur les méthodes classiques de résolution des équations aux dérivées partielles« : équations du premier ordre, caractéristiques, théorie de Hamilton Jacobi, classification des équations du second ordre, fonctions de Green, méthode de Riemann, etc. 

Prof. Félix Kwok

MAT 7225

Institution: Université Laval

Real Analysis and Measure Theory

Abstract theory of measure and integration: Borel-Cantelli lemmas, regularity of measures, product measures, Fubini-Tonelli theorem, signed measures, Hahn and Jordan decompositions, Radon-Nikodym theorem, differentiation in Rn.

Prof. Anush Tserunyan

MATH 564

Institution: McGill University

Mesure et intégration

Ensembles mesurables, mesure de Lebesgue, théorèmes de Lusin et de Egorov, intégrale de Lebesgue, théorème de Fubini, espaces Lp, éléments de la théorie ergodique, mesure et dimension de Hausdorff, ensembles fractals.

Prof. Maxime Fortier Bourque

MAT 6117

Institution: Université de Montréal

Équations aux dérivées partielles - Université de Montréal

Équations des ondes et de la chaleur, problème de Sturm-Liouville, théorie des distributions, espaces de Sobolev, fonctions harmoniques, équations elliptiques, éléments de la théorie spectrale.


 

Prof. Iosif Polterovich

MAT 6220

Institution: Université de Montréal

Winter

Topics in Analysis: Harmonic analysis and applications

The course will introduce students to the theory of classical harmonic analysis: convergence of Fourier series on the circle; Fourier transforms on the line and in Euclidean space; the Schwartz space and tempered distributions; and the Poisson Summation Formula. It will also cover applications to PDE; the Shannon Sampling Theorem; the discrete Fourier transform and Fast Fourier Transform; wavelets and frames.

 

Prof. Galia Dafni

MAST 661/4 sec. B/ 837B

Institution: Concordia University

Complex Analysis

The course is planned to consist of two parts: a short and condensed survey of the basic concepts of the theory of functions of one complex variable (from the Cauchy formula to the Riemann theorem on conformal mapping) and an introduction to the theory of compact Riemann surfaces (from elliptic functions to Abel and Riemann-Roch theorems; the latter will be introduced as a very special case of the index theorem).

Prof. Marco Bertola

MAST 665/4, sec. W /MAST 837W

Institution: Concordia University

Partial Differential Equations

The course is an introduction to the classical theory of partial differential equations (PDEs). The topics presented will be: first order linear and quasi-linear equations; linear second order PDEs (Laplace, Heat, Wave equations), maximum principles, properties of harmonic functions, accompanied by guided independent study, based on individual mathematical interests and areas of study, in which graduate students will explore further topics chosen from: nonlinear elliptic and parabolic PDEs (geometric properties of solutions, gradient flows, methods of subsolutions and supersolutions), or the use of calculus of variations and fixed point methods.

Prof. Alina Stancu

MAST 666/4 sec. A / 841A

Institution: Concordia University

Functional Analysis

Review of the basic theory of Banach and Hilbert spaces, Lp spaces, open mapping theorem,closed graph theorem, Banach-Steinhaus theorem, Hahn-Banach theorem, weak and weak-* convergence, weak convergence of measures, Riesz representation theorems, spectral theorem for compact self-adjoint operators, Fredholm theory, spectral theorem for bounded self-adjoint operators, Fourier series and integrals, additional topics.

Prof. Anush Tserunyan

MATH 565

Institution: McGill University

Analyse fonctionnelle avancée

Espaces de Sobolev. Algèbres de Banach, théorème de Gelfand. Théories spectrales d’opérateurs bornés. Opérateurs non bornés, transformée de Cayley.

Prof. Dmitry Faifman

MAT 6125

Institution: Université de Montréal

Analyse fonctionnelle (Sherbrooke)

Espaces de Hilbert, espaces de Banach, algèbres de Banach. Étude particulière de l'algèbre des opérateurs sur un espace de Hilbert. Espace de Banach des fonctions à variation bornée et intégrale de Stieltjes. Fonctionnelles linéaires. Théorème de représentation de Riesz. Théorèmes de Hahn-Banach, de la borne uniforme et du graphe fermé. Topologies faibles. Convexité : théorèmes de séparation, inégalité de Jensen, théorème de Krein-Milman.

Prof.

MAT 745

Institution: Université de Sherbrooke