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.

Given two polynomials with integer coefficients, for how many primes p do the polynomials induce nonisomorphic dynamical systems mod p? This question will lead us to the study of the statistics of wreath products, the Galois theory of dynatomic polynomials, and other topics. This work is joint with Andrew Bridy.

The framework of the Weil conjectures establishes a correspondence between the arithmetic of varieties over finite fields and the topology of the corresponding complex varieties. Many varieties of interest arise in sequences, and a natural extension of the Weil conjectures asks for a relationship between the asymptotic point count of the sequence over finite fields and the limiting topology of the sequence over C.  In this talk, I'll recall the Weil conjectures and explain the basic idea of these possible extensions.  I'll then give a survey of ongoing efforts to understand and exploit this relationship, including Ellenberg-Venkatesh-Westerland's proof of the Cohen-Leinstra heuristics for function fields, a ``best possible'' form of this relationship in the example of configuration spaces of varieties (joint with Benson Farb), and a counterexample to this principle coming from classical work of Borel and recent work of Lipnowski-Tsimerman. 

Let K be a local field with residue field of characteristic p>0. Our goal is to understand the cyclic extensions of K of degree a power of p. If K has characteristic 0 and contains a p^m-th primitive root of unity, then one can use class field theory and Kummer theory to construct a symbol which helps us to understand the ramification of cyclic extensions of degree p^m. If K has characteristic p, then one can construct a symbol, using class field theory and Artin-Schreier-Witt theory, which helps us to understand the cyclic extensions of degree p^m for any m. We will discuss both symbols in more detail and discuss methods for computing these symbols.