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.

While the sequence of primes is very well distributed in the reduced residue classes (mod q), the distribution of pairs of consecutive primes among the permissible pairs of reduced residue classes (mod q) is surprisingly erratic.  We propose a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures, which fits the observed data very well.  We also study the distribution of the terms predicted by the conjecture, which proves to be surprisingly subtle.  This is joint work with Kannan Soundararajan.

In the last several years, there has been significant theoretical progress on understanding the average rank of all elliptic curves over Q, ordered by height, led by work of Bhargava-Shankar. We will survey these results and the ideas behind them, as well as discuss generalizations in many directions (e.g., to other families of elliptic curves, higher genus curves, and higher-dimensional varieties) and some corollaries of these types of theorems. We will also describe recently collected data on ranks and Selmer groups of elliptic curves (joint work with J. Balakrishnan, N. Kaplan, S. Spicer, W. Stein, and J. Weigandt).