For $s>0$ and a lattice $L$ in $R^d$, we consider the asymptotics of $N$-point configurations minimizing the $L$-periodic Riesz $s$-energy as the number of points $N$ goes to infinity. In particular, we focus on the case $0<s<d$ of long-range potentials where we establish that the minimal energy $E_s(L,N)$ is of the form $E_s(L,N)=C_0 N^2 + C_1 N^{1+s/d} +o(N^{1+s/d})$ as $N\to \infty$ for constants $C_0$ and $C_1$ depending only on $s$, $d$, and the covolume of $L$. This is joint work with Ed Saff, Brian Simanek, and Yujian Su.

We investigate Positive Operator Valued Measures (POVMs) generated by Gabor frames in $\mathbb{C}^d$. A complete (Gabor) POVM is one that spans the space $\mathbb{C}^{d^{2}}$ of $d\times d$ matrices. It turns out that being a complete Gabor POVM is a generic property. As a result, the focus of this talk will be on non-complete Gabor POVMs. We will describe the possible ranks of these Gabor POVMs, and derive various consequences for the underlying Gabor frames. In particular, we will give details in dimensions $4$ and $5$.

We consider $L_2$-approximation of functions and want to compare the power of function values with the power of arbitrary linear information. Under mild assumptions on the class of functions, we show that the minimal worst-case errors based on function values decay at almost the same rate as those with arbitrary info, if the latter decay fast enough. Our results are to some extent best possible and, in special cases, improve upon well-studied point constructions, like sparse grids, which were previously assumed to be optimal. The proof is based on deep results on large random matrices, including the recent solution of the Kadison-Singer problem, and reveals that (classical) least-squares methods might be surprisingly powerful in a general setting.