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 topic of this talk will be understanding the p-adic slopes of modular forms. Recently, Bergdall and Pollack, based on computer calculations, raised a very interesting conjecture on the slopes of overconvergent modular forms, which predicts that the Newton polygons of the characteristic power series of U_p are the same as the Newton polygons of another explicit characteristic power series, which they call ghost series. This conjecture would imply many well-known conjectures regarding slopes of modular forms, like Gouvea's conjecture, Gouvea-Mazur conjecture, and etc. The goal of our joint project is to prove this conjecture under some mild hypothesis, and to explore some further application.  I will report on the progress so far.

Let P: ... -> C_2 -> C_1 -> P^1 be a Z_p-cover of the projective line over a finite field of characteristic p which ramifies at exactly one rational point. In this talk, we study the p-adic Newton slopes of L-functions associated to characters of the Galois group of P. It turns out that for covers P such that the genus of C_n is a quadratic polynomial in p^n for n large, the Newton slopes are uniformly distributed in the interval [0,1]. Furthermore, for a large class of such covers P, these slopes behave in an even more regular way. This is joint work with Hui June Zhu.

In this talk a lattice will mean a discrete subgroup Λ of n-dimensional Euclidean space; the sphere packing associated to Λ is the arrangement of congruent spheres of radius equal to one half the minimum distance of Λ and centered at the lattice points.  The main parameter under consideration will be the packing density of the arrangement of spheres.  With this in mind, a family of p-dimensional lattices will be constructed from submodules M of the ring of integers of a cyclic number filed L of degree p, where p is an odd unramified prime in L/Q.  The density of the associated sphere packing will be determined in terms of the nonzero minimum of the trace form in M and the discriminant of L.

In joint work with Daniel Hast, we recast the paper of Jon Keating and Zeev Rudnick "The variance of the number of prime polynomials in short intervals and in residue classes" by studying the geometry of these short intervals through an associated highly singular variety. We manage to recover their results for a a general class of arithmetic functions up to a constant and also obtain information about the higher moments. Recently work of Brad Rodgers in "Arithmetic functions in short intervals and the symmetric group" gives new insight into the geometry of our variety.