High dimensional geometry

High dimensional geometry
גיאומטריה רב ממדית

Fall semester,  2020/21
(October 25 till January 29)

 

  • Lecturer: Bo'az Klartag 
     
  • Teaching assistant: Yam Eitan
     
  • Classes:

    Wednesday, 14:15 - 16:00
    Thursday, 13:15 - 14:00

    Here is the zoom link.


    An exception: On the week of January 10-14, our schedule will be different. The lectures will be held on Tuesday, Wednesday and Thursday, 16:00 - 16:45, using another zoom link (part of a winter school at the Hausdorff Institute). 
     
  • Mailing list: Please fill this form in order to join the course mailing list. Students that plan to submit their homework should join the mailing list.
     
  • Syllabus

    Part I: High dimension 

    • Estimates for the central limit theorem for i.i.d random variables.
    • The isoperimetric inequality on the sphere, concentration of measure.
    • Maximal volume ellipsoid (John) and Dvoretzky's theorem: Any high-dimensional convex body contains approximately-spherical sections.
    • Thin shell theorem, Gaussian marginals with geometric assumptions on the random variables in place of independence.
       

    Part II: Convexity  

    • Brunn-Minkowski inequality, concentration for uniformly convex sets.
    • Volume-ratio and Kashin's theorem (approximately-spherical sections of almost full dimension).
    • Santalo inequality, Legendre transform, Brascamp-Lieb inequality.
    • The isotropic constant and the Bourgain-Milman inequality.
    • Milman ellipsoid, quotient of subspace, reverse Brunn-Minkowksi.
       
  • Prerequisites

    Familiarity with undergraduate probability, real analysis (say, Lebesgue measure) and Hilbert spaces.

  • Final grade

    It will be based on the solution of the homework exercises.
     

  • Related literature

    Recommended books from the 1980s:

    • Milman, Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces, 1986.

    • Pisier, The Volume of Convex Bodies and Banach Space Geometry, 1989.

    Recommended recent books on the subject:

    • Vershynin, High-Dimensional Probability, 2018.

    • Aubrun, Szarek, Alice and Bob meet Banach, 2017.

    • Artstein-Avidan, Giannopoulos, Milman, Asymptotic Geometric Analysis, 2015.

    • Brazitikos, Giannopoulos, Valettas, Vritsiou, Geometry of Isotropic Convex Bodies, 2014.

    • Boucheron, Lugosi, Massart, Concentration Inequalities, 2013.

    • Ledoux, The Concentration of Measure Phenomenon, 2001.

  • Lectures

    • The videos of all lectures should be available here, roughly one hour after the end of the lecture. Don't be misled by the black screen at the beginning of the video, the actual lecture starts afterwards.
       

    • Week 1: The high-dimensional cube and Euclidean ball, Central Limit Theorem. Here are class notes 1.

    • Week 2: Proof of the isoperimetric inequality on the sphere, a bit about concentration of measure phenomenon. Here are class notes 2.

    • Week 3: Concentration of Lipschitz functions, proof of the thin shell theorem. Here are class notes 3, plus a tutorial on Brunn-Minkowski.

    • Week 4: Applications of concentration phenomena, Johnson-Lindenstrauss lemma, John's ellipsoid. Here are class notes 4.

    • Week 5: A proof of Dvoretzky's theorem. Here are class notes 5.

    • Week 6: Volume/diameter balance of sections, Kashin's decomposition and finite volume ratio, Grothendieck's inequality. Here are class notes 6.

    • Week 7: Dudley's inequality, removal of log(epsilon) from Dvoretzky's theorem, statements of Sudakov's inequality, low M* estimate and Talagrand's theorem. Here are class notes 7.

    • Week 8: Gaussian concentration (Maurey-Pisier proof), log-concave densities, Prekopa-Leindler, reverse Hölder inequalities of Berwald-Borell. Here are class notes 8.

    • Week 9: Concentration for uniformly convex bodies, entropy and covariance of log-concave measures, isotropic constants. Here are class notes 9.

    • Week 10: The slicing problem and preparation for the winter school next week, mostly stochastic processes. Here are class notes 10.

    • Week 11: Yuansi Chen's work on the KLS conjecture (part of the winter school at the Hausdorff institute). Here are class notes 11, typed in latex, as well as the traditional handwritten version. The videos are here: lecture 1, lecture 2 and lecture 3.

    • Week 12: Proof of the Bourgain-Milman inequality and the existence of M-ellipsoid. Here are class notes 12.

    • Week 13: Reverse Brunn-Minkowski, quotient of subspace thm, and other application of M-position. Here are class notes 13.

  • Assignments

English