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Energy Optimization for k-particle Interactions on the Sphere

Ryan Matzke, Vanderbilt University
Tuesday, April 11, 2023 - 1:30pm to 3:30pm
PDL C-401

A great deal of study has been devoted to energies involving two-particle interactions. Such systems have applications in discrete geometry, signal processing, and modeling various natural phenomena (e.g., electrostatic or gravitational energy), among other uses. Energies involving many-particle interactions are far less understood, which have been appearing recently to improve upon previous results in discrete geometry (such as bounds for optimal codes and kissing numbers) which made use of two-particle interactions. In this talk, we will discuss recent work in developing general theory for energy optimization on the sphere for multivariate potentials (i.e. ones that model many-particle interactions), some connections and applications, and a variety of open problems. In particular, we will discuss geometric kernels, such powers of the volume of a simplex generated by k points (a multivariate generalization of Riesz kernels). The work in this talk covers work done in collaboration with Dmitriy Bilyk (University of Minnesota), Damir Ferizovic (KU Leuven), Alexey Glazyrin (University of Texas Rio Grande Valley), Josiah Park (Texas A&M University), and Oleksandr Vlasiuk (Vanderbilt University).

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