Volumes in High Dimension
נפחים בממד גבוה
Spring semester 2018
March 15, 2018  July 6, 2018
 Lecturers
Bo'az Klartag and Uri Grupel
 Classes
Uri: Wednesday, 14:15  15:00, Ziskind 155
Bo'az: Thursday, 10:45  11:30, Ziskind 1 and 11:45  12:30, Ziskind 155
 Course Description
A priori, one would expect geometry in highdimensional spaces  even in Euclidean spaces  to be rather complicated. Our experience in two and three dimensions seems to indicate that as the number of dimensions increases, the number of possible configurations grows rapidly, and we enter the realm of enormous, unimaginable diversity.
Nevertheless, in this class we will see that dimensionality, when correctly viewed, may become a blessing. There are motifs in highdimensional geometry, most notably the concentration of measure, which seem to compensate for the vast amount of different possibilities. Convexity is one of the ways in which to harness this and other motifs and thereby formulate clean, nontrivial theorems.
We plan to cover most of the following 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 highdimensional convex body contains approximatelyspherical sections.
 Analytic techniques: Poincare and LogSobolev inequalities, martingales, curvature.
 Thin shell theorem, Gaussian marginals with geometric assumptions on the random variables in place of independence.
Part II: Convexity
 BrunnMinkowski inequality, concentration for uniformly convex sets.
 Volumeratio and Kashin's theorem (approximatelyspherical sections of almost full dimension).
 Slepian's lemma and Sudakov's inequality
 Low M*estimate, JohnsonLindenstrauss
 Complex interpolation and Kconvexity.
 Milman ellipsoid, quotient of subspace, Santalo & reverseSantalo, reverse BrunnMinkowksi.
 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
There are a few books covering this material, here are some recommended ones:

ArtsteinAvidan, Giannopoulos, Milman, Asymptotic Geometric Analysis, part I, 2015.

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

Boucheron, Lugosi, Massart, Concentration inequalities, 2013.

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

Ledoux, The concentration of measure phenomenon, 2001.

Milman, Schechtman, Asymptotic theory of finitedimensional normed spaces, 1986.

Pisier, The volume of convex bodies and Banach space geometry, 1989.

Vershynin, HighDimensional Probability. The book hasn't appeared yet, a draft is available here.

 Comments
 An argument related to Slepian's lemma.
 An argument related to Slepian's lemma.
 Homework (written by Uri)
"Il serait paradoxical que le grand nombre des variables fût une cause de simplicité", P. Levy, Problemes concrets d'analyse fonctionnelle, page xi.
(Free translation: It is paradoxical that simplicity could arise from a large number of variables)