25 April 2019: Kolmogorov Day

Abstract: If a tree is chosen at random among all trees with n vertices, what is the typical distance between a pair of vertices? What is the length of the longest increasing subsequence in a random permutation? What is the chance that the random power series with i.i.d. complex Gaussian coefficients has no zeros in the disk of radius 1/2? These questions seem unrelated, but the common framework of determinantal points processes provides answers to all of them. Determinantal point processes are a class of random discrete sets with a specific form of dependence between points. The definition is motivated by the idea of non-interacting fermions in Quantum physics, but the real motivation is that there are innumerable examples in probability, combinatorics and mathematical physics and that these processes share many common properties. Some examples are the uniform random spanning tree on a graph, the eigenvalues of certain random matrices, zeros of certain random power series, etc. There are also applications to sampling problems in theoretical computer science and possible models (in place of the usual Poisson model) for sensor networks in communication theory. We shall give a survey of this area, focusing on examples. The lecture is aimed to be accessible to anyone with a graduate level understanding of analysis and probability.