Discrete and continuous energy problems that arise in a variety of scientific contexts are introduced, along with their fundamental existence and uniqueness results. Particular emphasis will be on Riesz and Gaussian pair potentials and their connections with best-packing and the discretization of manifolds. The latter application leads to the asymptotic theory (as N → ∞) for N-point configurations that minimize energy when the potential is hypersingular (short-range). For fixed N, the determination of such minimizing configurations on the d-dimensional unit sphere S d is especially significant in a range of contexts that include coding theory, discrete geometry, and physics. We will review linear programming methods for proving the optimality of configurations on S d , including Cohn and Kumar’s theory of universal optimality. The following reference will be made available during the short course: Discrete Energy on Rectifiable Sets, by S. Borodachov, D.P. Hardin and E.B. Saff, Springer Monographs in Mathematics, 2019.

Sessions:

Friday, 24 January 14:00-16:00

Friday, 31 January 14:00-16:00

We investigate the strong data processing inequalities of contractive Markov Kernels under a specific f-divergence, namely the E-gamma-divergence. More specifically, we characterize an upper bound on the E-gamma-divergence between PK and QK, the output distributions of contractive Markov kernel K, in terms of the E-gamma-divergence between the corresponding input distributions P and Q. Interestingly, the tightest such upper bound turns out to have a non-multiplicative form. We apply our results to derive new bounds for the local differential privacy guarantees offered by the sequential application of a privacy mechanism to data and we demonstrate that our framework unifies the analysis of mixing times for contractive Markov kernels.