Title: Energy-minimizing measures and point configurations on the sphere.

Abstract: Numerous problems, arising in discrete and metric geometry, signal processing, physics, etc, can be reformulated as questions of optimizing discrete or continuous energies. We shall discuss some problems and conjectures of this type, as well as their connections to various topics and objects, such as discrepancy, sphere packings, spherical designs, optimal codes, tight frames, equiangular lines, mutually unbiased bases, etc. While in many natural examples, energy-minimizing measures or point configurations tend to be uniformly distributed, a peculiar effect is observed for certain energy functionals, especially with attractive-repulsive potentials: minimizing measures are not spread out over the domain, but rather turn out to be discrete (or supported on very thin sets). We shall also explore this phenomenon. The majority of examples in the talk will deal with energies on the sphere.

About the Speaker: Dimitriy Bilyk originally from Ukraine, received his PhD in 2005 from the University of Missouri at Columbia under the supervision of Loukas Grafakos. Bilyk was first a Visiting Assistant Professor at Georgia Institute of Technology, a Visiting Member of both the Fields Institute in Toronto, Canada, and the Institute for Advanced Study in Princeton. In 2008 he joined the University of South Carolina for four years moving in 2012 to the University of Minnesotta  where he is a Full Professor. His research interests are in Harmonic and Functional Analysis, Discrepancy Theory, Approximation, and Probability. He has been a recipient of several NSF research grants as well as conference grants. He has mentored and collaborated with  many graduate students and also undergraduate students.

Dima came to NM at least twice for joint NM Analysis Seminars paired with AMS meetings (2007, 20014).