Codes and Expansions (CodEx) Seminar

Ryan Matzke (Vanderbilt University):
Energy Minimization: Concentration and Dispersion of Charge

Finding optimal point distributions is a problem that appears in several fields, such as numerical integration, coding theory, and chemistry, among others. One way of determining “good” point distributions is through the minimization of discrete energy functionals \[ E_K(\omega _N) = \sum _{i \neq j} K(z_i, z_j)\] over \(N\) point sets \(\omega _N := \{ z_1, ..., z_N\} \subset \Omega \subseteq \mathbb {R}^d\), or their continuous counterparts \[I_K(\mu ) = \int _{\Omega } \int _{\Omega } K(x,y) d\mu (x) d\mu (y),\] over probability measures \(\mu \in \mathcal {P}(\Omega )\).

In this talk, we will consider the \(p\)-frame energy, with kernel \(K(x,y) = |\langle x ,y \rangle |^p\) for \(p \in (0, \infty )\), and \(\Omega \) the unit sphere, discussing what is currently known about minimizers for these energies, as well as their connections to discrete geometry, quantum information theory, and convex geometry.