Abstracts for Summer School Lectures

 

Zakhar Kabluchko (University of Münster): Random Polytopes

A polytope is a convex hull of finitely many points in Euclidean space. By taking these points to be random, we obtain random polytopes. Examples include convex hulls of independent identically distributed random points (including the so-called Gaussian polytope which arises if the points have standard Gaussian distribution), convex hulls of multidimensional random walks, random projections of regular polytopes, and many others. We shall be interested in computing expectations of various functionals of such polytopes, for example the volume, the number of faces, internal and external angles, and some others. It turns out that there are many beautiful interrelations between these functionals. For example, Baryshnikov and Vitale observed that the number of faces has the same distribution for Gaussian polytopes as for projections of regular polytopes. The main tool used in our computations is the integral geometry of convex cones. We shall introduce the participants to this subject. In particular, we shall give various definitions of intrinsic volumes for convex cones.  Also, we shall address some problems of classical geometry.  For example, we shall compute the number of parts in which $n$ hyperplanes in general position divide the $d$-dimensional space. Surpsingly, this problem is equivalent to the following one: compute the probability that the Gaussian polytope contains the origin. Finally, we shall provide some applications of conic integral geometry to problems of high-dimensional statistics and convex optimization in the spirit of the paper by Amelunxen, Lotz, McCoy and Tropp https://arxiv.org/abs/1303.6672

 

Ashkan Nikeghbali (University of Zurich): Random Phenomena, Primes and Zeta 

Random models can be used to predict the asymptotic  behaviour of some arithmetic functions; but they might also hold to predict their finer behaviours.  In this lecture, we will use two examples of arithmetic functions and corresponding probabilistic models to illustrate this fact. We shall cover limit theorems at various scales  as well as some exact finite N computations related to random unitary matrices, which is also of independent interest. The goal of these lectures is to emphasise probabilistic methods.

 

Sandrine Péché (Université Paris Diderot): Deformed and Exactly Solvable Models of Random Matrices

In this course we will discuss some famous ensembles of deformed random matrices focusing on those for which exact computations can be performed. A detailed analysis of the spectral properties of such random matrices will be given. Extensions to more general ensembles will also be  adressed.