Integrable Systems and Random Matrix Theory Seminar

How are holes in the pollen particles distributed in such a way as to optimize germination? How do we determine the organization of the proteins that cover the viruses? What is the optimal encapsulation of active ingredients such as drugs, nutrients or living cells? All of these structures correspond to an intuitive notion of "well distributed points" in the two dimensional sphere. A question arises naturally: what is the explicit mathematical definition of well distributed points? The answer varies a lot: from points that minimize the error of interpolation to best packing problems passing through points that minimize a potential on the sphere. During this talk we will explore the configurations of points in d-dimensional spheres that minimize some potential or energy. Surprisingly, some of the best configurations in this sense are random configurations of points. In particular, they come from the so-called Determinantal Point Processes. We will see some properties of Determinantal Point Processes and how we can take advantage of them in order to obtain well distributed points on the sphere. Speaker(s): Uju\'e Etayo (Universidad de Cantabria (Spain))