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)

2023-24 Course Listings

Fall

Functional Analysis II

The course is devoted to the theory of unbounded operators in Hilbert spaces. The main themes are extensions of symmetric operators and criteria of self-adjointness, proofs of the spectral theorem for unbounded operators, applications to PDE. As an additional topic I am planning to include some versions of the adiabatic theorem for time-dependent Hamiltonians and elements of Berry phase theory.        

Prof. A. Kokotov

MAST 661E / MAST837E

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. Anush Tserunyan

MATH 564

Institution: McGill University

Advanced Topics in Analysis

Prof. Pengfei Guan

MATH 740

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

Analyse fonctionnelle (UdeM)

Espaces 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. Iosif Polterovich

MAT 6124

Institution: Université de Montréal

Analyse fonctionnelle I

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. Tomasz Kaczynski

MAT 745

Institution: Université de Sherbrooke

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; applications to sampling theory; the discrete Fourier transform and Fast Fourier Transform; wavelets and frames.

Students will be able to choose further topics from theoretical or applied harmonic analysis to pursue in individual projects/presentations.

Prof. Galia Dafni

MAST 661 / MAST 837

Institution: Concordia University

Partial Differential Equations

Linear and quasilinear 1-st order equations. Transport equation. Shock waves and rarefactions.  D'Alembert solution to the one-dimensional wave equation. Infinite, semiinfinite and finite string.  Separation of variables, Fourier method for the 1-d wave equation.  Solution of the wave equation in 2-d and 3-d. Duhamel formula. Energy method, finite speed of propagation.  Laplace and Poisson equations in 2-d and 3-d. Green's formula. Hydrodynamical interpretation.  Properties of harmonic functions. Maximum principle, mean value theorem, Liouville and Harnack's theorems.  Dirichlet's and Neumann's problems for the Laplace equation. Variational method.  Heat equation. Solution in the whole space. Energy method for the proof of existence and uniqueness of solution.

Prof. Alina Stancu

MAST 666 / MAST 841

Institution: Concordia University

Measure Theory

Measure and integration, measure spaces, convergence theorems, Radon-Nikodem theorem, measure and outer measure, extension theorem, product measures, Hausdorf measure, LP-spaces, Riesz theorem, bounded linear functionals on C(X), conditional expectations and martingales.

Prof.

MAST 669 / MAST 837

Institution: Concordia University

Variétés et formes différentielles

Formes différentielles dans l'espace euclidien. Variétés différentielles. Intégration de formes différentielles, théorème de Stokes, applications.

Prof. Thomas Ransford

MAT 7155

Institution: Université Laval

Advanced Real Analysis 2

Continuation of topics from MATH 564. Signed measures, Hahn and Jordan decompositions. Radon-Nikodym theorems, complex measures, differentiation in Rn, Fourier series and integrals, additional topics.

Prof. John Toth

MATH 565

Institution: McGill University