This is an introductory online school for the EIMI thematic program “New Trends in Mathematical Stochastics” to be held in St. Petersburg in 2021 (rescheduled from 2020 due to COVID-19 pandemic). The aim of the school is to introduce young researchers to several important topics in stochastic geometry and probability theory including random polytopes, random matrices, and floating bodies. Senior researchers interested in learning these topics are also welcome.
The school consists of 3 courses 4 lectures each. The lecture courses are:
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 unexpected 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. As shown by Wendel, this problem is equivalent to the following one: compute the probability that the Gaussian polytope contains the origin.
Random polytopes, random cones, and their integral geometric characteristics appear naturally in the Grassmannian approach to linear programming suggested by A. M. Vershik. We shall provide some examples following the paper by Amelunxen, Lotz, McCoy and Tropp https://arxiv.org/abs/1303.6672
This lecture series will be about the use of tridiagonal random matrices in the study of the log gas on the real line (particles on the line in a quadratic potential well, repelling by logarithmic interaction energy).
Starting with the innovative ideas of Trotter and then Dumitriu-Edelman who found these models, we shall cover some of the basic results such as Selberg integral, Wigner's semicircle law, Furedi-Komlos and then go on to study the behaviour of particles at the edge, moderate deviations for the largest eigenvalue, the beta Tracy-Widom distributions of Ramirez-Rider-Virag via operator limits.
Other related topics that may appear are Szego's theorem on asymptotics of zeros of orthogonal polynomials and Kerov's theorem on separation of interlaced roots of orthogonal polynomials. In the end, we hope to talk about some recent works on law of iterated logarithm in exponential last passage percolation (due to Ledoux and Basu–Ganguly–Hegde–K.), using the results from earlier lectures and a well-known mapping of LPP with Wishart random matrices.
Overall, it is expository in nature and aimed at graduate students and does not assume prior knowledge of random matrix theory.
1) Holcomb, D. and Virag, B., Operator limits of random matrices, https://people.kth.se/~holcomb/ShortCourseNotes.pdf
2) Krishnapur, M., Random matrix theory (notes from a course), http://math.iisc.ac.in/~manju/RMT17/RMT_2017.pdf
Two important closely related notions in affine convex geometry are the floating body and the affine surface area of a convex body.
The floating body of a convex body is obtained by cutting off caps of volume less or equal to a fixed positive constant. Taking the right-derivative of the volume of the floating body gives rise to an affine invariant, the affine surface area. This was established for all convex bodies in all dimensions by Schuett and Werner. There is a natural inequality associated with affine surface area, the affine isoperimetric inequality, which states that among all convex bodies, with fixed volume, affine surface area is maximized for ellipsoids.
Due to its important properties, which make them effective and powerful tools, affine surface area and floating body are omnipresent in geometry and have applications in many other areas of mathematics, e.g., in problems of approximation of convex bodies by polytopes and for the notion of halfspace depth for multivariate data from statistics.