A priori, one would expect geometry in high-dimensional 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 high-dimensional 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, non-trivial 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 high-dimensional convex body contains approximately-spherical sections.
- Analytic techniques: Poincare and Log-Sobolev inequalities, martingales, curvature.
- 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).
- Slepian's lemma and Sudakov's inequality
- Low M*-estimate, Johnson-Lindenstrauss
- Complex interpolation and K-convexity.
- Milman ellipsoid, quotient of subspace, Santalo & reverse-Santalo, reverse Brunn-Minkowksi.
"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)