Phénomènes de grande dimension (M2), 2019

Les cours ont lieu le lundi en salle Séminaire 1 de 13h à 16h15 (en sous-sol du bâtiment Braconnier, l'accès nécessite un badge, rendez-vous à 13h au pied de l'escalier extérieur ouest).

  • Lesson 1 (14 janvier). Convexity : the Brunn-Minkowski theory. Basic notions on convex bodies. Prékopa-Leindler and Brunn-Minkowski inequalities. Isoperimetric inequality. Polar of a convex body. Blaschke-Santalo inequality (symmetric case, proof by Steiner symmetrisation).

  • Lesson 2 (21 janvier). The Banach-Mazur compactum. Banach-Mazur distance. John's theorem, caracterization of the maximal volume ellipsoid by contact points. Compactness of the Banach-Mazur compactum. Distance between B_2^n and B_1^n. Distance between B_1^n and B_{inf}^n ; Hadamard matrices, Khintchine inequalities.

  • Lesson 3 (28 janvier). The concentration of measure phenomenon. Uniform measure on the sphere, estimates on the measure of a spherical cap. Packing and covering on the sphere ; random covering argument. Isoperimtric inequality on the sphere, and measure concentration (Lévy's lemma) as a corollary. Isoperimetry and concentration in the Gaussian space (derived from the spherical case). Application : Johnson-Lindenstrauss lemma.

  • Lesson 4 (4 février). Dvoretzky's theorem. Warm-up: isometric embedding of L^2 in L^p via Gaussian variables. The goal of the course is to prove Dvoretzky(-Milman) theorem. Special case: almost Euclidean section of the cube. Haar measure on O(n), on the Grassmann manifold. Concentration of Lipschitz function on subspaces. Estimates on the average of norms of the sphere via Gaussian variables ; Dvoretzky-Rogers lemma.

  • Lesson 5 (11 février). More on volume of convex bodies Warm-up: volume and volume radius of B_1^n and B_{\iy}^n. The volume ratio theorem. Application : Kashin decomposition of L^1. Polytopes ; mean width, Urysohn inequality, upper bound on volume of polytopes. Vertex/facet complexity of symmetric polytopes via Dvoretzky's theorem.

  • Lesson 6 (25 février). Gluskin's theorem Preliminary : volume of the operator norm unit ball ; rough estimates on norms of Gaussian random matrices. Complete proof of Gluskin's theorem on the diamater of the Banach-Mazur compatcum. Slepian's lemma (proof postponed) and application to Gaussian random matrices.

  • Lesson 7 (4 mars). Gaussian processes Proof of Slepian and Gordon's lemma. Covering numbers : Sudakov inequality, dual Sudakov inequality, Dudley inequality. Definition of the ell-norm and the ell-position of a symmetric convex body.

  • Lesson 8 (11 mars). ell-ellipsoid. Mean width estimates in ell-position. Hermite polynomials and Ornstein-Uhlenbeck semigroup : definitions and basic properties. Definition of the K-convexity constant. Proof of the bound log(n) for the K-convexity constant of a n-dimensional convex body.

  • Lesson 9 (18 mars). M-ellipsoid and related results. Reverse Santalo inequality : proof via isomorphic symmetrizations. Existence of the M-ellispoid. Corollary : reverse Brunn-Minkoswki inequality. Quotient of subspace theorem.

    Exercise sheets

  • Sheet #1(in French; next sheets are in English)
  • Sheet #2
  • Sheet #3
  • Sheet #4
  • Sheet #5
  • Sheet #6
  • Sheet #7
  • Sheet #8

    Final exam

    Here is the exam and solutions

    Références

  • Roman Vershynin, High-dimensional Probability
  • Guillaume Aubrun et Stanislaw Szarek, Alice and Bob meet Banach
  • Shiri Artstein-Avidan, Apostolos Giannopoulos et Vitali Milman, Asymptotic Geometric Analysis (Part I)
  • Daniel Li et Hervé Quéffélec, Introduction à l'étude des espaces de Banach