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.
We will show that if a discrete d-dimensional Schr\"odinger operator has only discrete spectrum outside the interval [-2d, 2d], then its essential spectrum must be [-2d, 2d].
This talk will be based on a recent work by Killip, Molchanov and Safronov.
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.
The Southern California Applied Mathematics Symposium (SOCAMS) will be held on Saturday, June 3rd at UC Irvine. SOCAMS is a one-day meeting, featuring talks by applied mathematicians from Southern California.
We will complete our discussion of the quantum algorithm to
compute the unit group of a number field. We will then discuss
applications by Biasse and Song to compute class groups and generators
of principal ideals. The paper of Biasse and Song is available on my
webpage,