Abstract: Semiclassical resolvent norms relate dynamics of a particle scattering problem to regularity and decay of waves in a corresponding wave scattering problem. In my talk I will discuss the effect that geometric trapping of particles has on resolvent norms. I will focus in particular on the phenomena of propagation of singularities and quantum tunneling, in the setting of scattering by a compactly supported smooth function in Euclidean space. This talk is based in part on joint works with Long Jin and Andras Vasy.

To any point p on a smooth algebraic curve C, the Weierstass semigroup is the set of all possible pole orders at p of regular functions on C \ {p}. The question of which sets of integers arise as Weierstass semigroups is a very old question, still widely open. We will describe progress on the question, defining a quantity called the effective weight of a numerical semigroup, and describe a proof that all numerical semigroups of sufficiently small effective weight arise as Weierstrass semigroups. The proof is based on older work of Eisenbud, Harris, and Komeda, based on deformation of certain nodal curves. We will survey some combinatorial aspects of the effective weight, and various open questions regarding both numerical semigroups and algebraic curves.