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:
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 properties of convex sets and functions, the course aims to present several classical inequalities like the Brunn-Minkowski inequality and its related functional form, Prekopa-Leindler, the Blaschke-Santaló inequality, the Urysohn inequality, as well as more recent results such as the reverse isoperimetric inequality, and 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.
The main parts of the course will consist of the following topics: Set theory; the real number system; Metric spaces; Topological spaces; Compact spaces; Banach spaces.
In the first part of the course, we will develop the method of forcing and prove the independence of the Continuum Hypothesis from ZFC, and the independence of the Axiom of Choice from ZF. We will cover several technical notions which are crucial for modern day set theory, including: chain conditions; closure and distributivity conditions; product forcing and mutual genericity; collapse forcing; projections and isomorphism of forcing notions.
Given time, we will continue to more advanced applications of forcing, in particular to descriptive set theory. We will start with proving the consistency of ZF + DC + 'all sets of reals are Lebesgue measurable', assuming the existence of an inaccessible cardinal (the Solovay Model).
Measure and integration, measure spaces, convergence theorems, Radon-Nikodym theorem, measure and outer measure, extension theorem, product measures, Hausdorff measure, Lp spaces, Riesz theorem, bounded linear functionals on C(X), conditional expectations and martingales.
La géométrie spectrale est l'étude des liens entre la géométrie d'un espace et les valeurs propres d'opérateurs naturellement dénis sur celui-ci. Dans ce cours nous étudierons des opérateurs de type Laplacien et Dirichlet-Neumann sur des espaces tels que des surfaces et des domaines de l'espace euclidien. L'accent sera mis sur les inégalités géométriques et les méthodes variationnelles permettant de les étudier.
La première partie du cours sera consacrée à l'étude de sujets classiques en géométrie spectrale: calcul des valeurs propres pour des exemples simples (rectangles, disques, sphères, etc); théorème spectral pour le Laplacien et l'opérateur de Dirichlet-Neumann; caractérisation variationnelle des valeurs propres; asymptotique spectrale; géométrie nodale des fonctions propres; optimisation de forme sous contrainte de type isopérimétrique pour l'écart spectral λ1 des surfaces et pour les domaines du plan; inégalités géométriques en dimension arbitraire.
Par la suite, des perspectives et des thèmes récents seront abordés: homogénéisation en optimisation spectrale; valeurs propres des graphes et discrétisation des problèmes spectraux; perturbations et continuité des valeurs propres; construction d'espaces dont l'écart spectral λ1 est arbitrairement grand, méthode par capaciteurs pour les valeurs propres λk d'indice arbitraire, etc. Nous les choisirons ensemble en fonction de vos intérêts et connaissances.
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.
Topics in classical descriptive set theory concerning Polish spaces, regularity properties of sets such as measurability/Baire measurability and their connection with infinite games (determinacy), the Borel and projective sets/hierarchies, and change of topology techniques, as well as more modern topics on definable equivalence relations and classification, Polish group actions, and graph combinatorics on Polish spaces.
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.
Le laplacien et la théorie elliptique. Espaces de Sobolev. Éléments de la géométrie spectrale. Applications analytiques et topologiques à la géométrie riemannienne, symplectique ou kahlerienne.
The course will consist of the following topics, with possible additions if time permits:
Hilbert spaces, Banach spaces, linear functional dual spaces, bounded linear operators, adjoints, The Hahn-Banach theorem, Baire category theorem, Banach-Steinhaus theorem, open mapping and closed graph theorems, compact operators, the spectral theorem for self-adjoint compact operators, the Fredholm alternative, the weak/weak* topological vector spaces, distributions, Sobolev spaces.
Students will be required to complete an independent study project on a topic of their choice as approved by the instructor, related to the course material, and submit it as a written report and in the form of an oral presentation.
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.
Suggested references: Partial Differential Equations: A First Course, Rustum Choksi (2022) Partial Differential Equations, by Lawrence C. Evans (2010).
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.
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.
Théorie abstraite de l'intégration. Mesures de Borel et théorème de représentation de Riesz. Espaces Lp. Mesures complexes et théorème de Radon-Nikodym. Intégration sur les espaces produits et le théorème de Fubini. Différentiation.