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 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.
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.
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.
We shall discuss the following topics (as time permits). Fourier series: summability in norm, pointwise convergence, order of magnitude of Fourier coefficients, L2 theory, absolutely convergent Fourier series, convergence in norm, convergence and divergence at a point, sets of divergence (if time permits). Interpolation: Riesz-Thorin and Hausdor-Young theorems. Lacunary series. Fourier transforms: FT for L1; L2; Lp, tempered distributions, almost-periodic functions, Payley-Wiener theorems.
Possible additional topics (some of which could be used for presentation): Wirtinger's inequality, Isoperimetric inequality, Poisson summation formula, applications to number theory (e.g. theta functions and Gaussian sums), compact operators, applications to probability, applications to PDE (e.g. solving heat and wave equations), Heisenberg uncertainty principle, harmonic analysis on Abelian groups, representation theory, random Fourier series.
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.
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.
While graphs are intuitively and naturally represented by vertices and edges, such representations are limited in terms of their analysis, both theoretically and practically (e.g., when implementing graph algorithms). A more powerful approach is yielded by representing them via appropriate matrices (e.g., adjacency, diffusion kernels, or graph Laplacians) that capture intrinsic relations between vertices over the "geometry" represented by the graph structure. Spectral graph theory leverages such matrices, and in particular their spectral and eigendecompositions, to study the properties of graphs and their underlying intrinsic structure. This study leads to surprising and elegant results, not only from a mathematical standpoint, but also in practice with tractable implementations used, e.g., in clustering, visualization, dimensionality reduction, and manifold learning, and geometric deep learning. Finally, since nearly any modern data nowadays can be modelled as a graph, either naturally (e.g., social networks) or via appropriate affinity measures, and therefore the notions and tools studied in this course provide a powerful framework for capturing and understanding data geometry in general.
Accessibilité de langue : The course will accommodate anglophone students who are interested in taking it, as well as francophone students.
On utilisera la mécanique classique et le principe de moindre action pour s'initier aux concepts de base du calcul des variations, notamment les équations d'Euler-Lagrange et les équations d'Hamilton. On transposera alors ces notions en géométrie en abordant plusieurs exemples intéressants: géodésiques, surfaces minimales, métriques à courbure constante, applications harmoniques, flot gradient, théorie de jauge. On se concentrera alors sur les surfaces minimales en s'initiant à une méthode systématique pour les construire: la théorie géométrique de la mesure.
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.
Topics include: Hilbert spaces, Banach spaces, linear functionals, dual spaces, bounded linear operators, adjoints; the Hahn-Banach, Baire caterogy, Banach-Steinhaus, open mapping and closed graph theorems; compact operators, the Fredholm alternative, the spectral theorem; the weak/weak* topologies.
Mathematical logic studies mathematical objects by formalizing them in a precise “mathematical language” and then studying how these objects can be defined (or expressed) in this language.
The following concepts will be covered: mathematical structure, isomorphism, logical implication, formal deduction, countable and uncountable sets, Peano Arithmetic, computable set, computable function. The key results which will be covered are: The Completeness Theorem, The Compactness Theorem, Cantor’s Theorem, Godel’s Incompleteness Theorem.
The syllabus can be found here: https://sites.google.com/view/assaf-shani/teaching
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.
The first part of the course will be on Riemann surfaces (definition, examples, holomorphic functions and their properties, the uniformization theorem and its consequences, hyperbolic geometry) and the second part will be on the theory of their deformations (Fenchel-Nielsen coordinates, quasiconformal maps, extremal length, Beltrami and quadratic differentials, Teichmüller's theorem).
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.
Ce cours est une introduction à la théorie des surfaces de Riemann. Le préalable exigé est une connaissance de base de l'analyse complexe.
Surfaces de Riemann compactes. Structures complexes engendrées par une métrique. Applications holomorphes. Revêtements ramifiés de la sphère de Riemann, formule de Riemann-Hurwitz. Topologie et formes différentielles sur les surfaces de Riemann. Différentielles abéliennes, Jacobien. Fonctions méromorphes sur les surfaces de Riemann compactes. Théorème d'Abel. Théorème de Riemann-Roch. Fonctions théta, fonctions de Weierstrass. Aperçu des courbes algébriques.