Time: 16:15-17:45 (TBC)

Location: Mondi 3

Speaker 1: Yunzhe Li (Kaloshin group)
Title: Hearing the shape of a polygon

Abstract: Consider an ideal billiard ball moving inside a polygonal table. The billiard trajectory can be coded by the sequence of edges hit by the ball. The bounce spectrum of a polygon is defined as the set of bi-infinite sequences arising from the coding of billiard trajectories in the polygon. I will discuss whether one can reconstruct the polygon from its bounce spectrum. This question can be viewed as a symbolic analogue of the question ‘can you hear the shape of the drum’. After reviewing existing results, I will explain a basic theorem in this direction by Galperin, Kruger, and Troubetzkoy, showing that non-periodic billiard orbits can be 'heard'.

Speaker 2: Filippo Quattrocchi (Maas group)
Title: Quantization of probability measures

Abstract: How, and how well, can we approximate a measure with a discrete one, i.e., with a given number n of supporting points? This question arises naturally in many scientific fields: information theory, data science, numerical analysis, economics, etc. I will introduce the fundamentals of the theory of quantization (or discretization) of measures and describe Zador's theorem: a classical statement on the asymptotic behavior of the approximation error as n tends to infinity. I will also present some new results on a "uniform" variant of the problem: what if the approximating measure has to be uniform?