Wednesday 14 May 2025, at 11 am in Room 3014 (3rd Floor) - U5 Building, Prof. Dmitriy Bilyk (University of Minnesota) will give the following talk
Title: Energy minimization problems in analysis and discrete geometry
Abstract: We discuss problems of minimizing discrete and continuous pairwise interaction energies, i.e. expressions of the form ∑i≠jK(xi,xj) and ∫Ω∫ΩK(x,y)dμ(x)dμ(y),\sum_{i\neq j} K(x_i, x_j) \text{ and } \int_\Omega \int_\Omega K(x,y) d\mu (x) d \mu(y),where x1,…xnx_1, \dots x_n are points in a given domain Ω\Omega and μ\mu is a probability measure on Ω\Omega. Minimizing such energies can be interpreted as finding the optimal distribution of n particles or of the continuous unit charge under the interaction defined by the kernel K. Such problems naturally arise in numerous areas of mathematics: discrete and metric geometry, analysis, potential theory, signal processing, geometric measure theory, discrepancy theory, mathematical physics etc. In many classical settings, the minimizing distribution is in some sense uniform, however, a peculiar effect can be observed for some energies: minimizers are discrete or are supported on small sets, i.e. optimal distributions tend to cluster. We shall discuss a variety of problems and energies (sums of distances, Riesz energies, p-frame energies) with a particular focus on this mysterious clustering phenomenon.
All interested are invited to participate