If E and E' are elliptic curves defined over a finite field k such that E(k) and E'(k) have the same order, what is the likelihood that they define isomorphic groups?  In this talk we will address this question from two points of view: fix E, vary p, and fix p, vary E.  This is recent and ongoing work with Nathan Kaplan of UC Irvine.

One of the central problems in Arithmetic Statistics is counting number field extensions of a fixed degree with a given Galois group, parameterized by discriminants. We will focus on C₂≀H extensions over an arbitrary base field. While Jürgen Klüners has established the main term in this setting, we present an alternative approach that provides improved power-saving error terms for the counting function.

The Bateman--Horn Conjecture predicts how often an irreducible polynomial  assumes prime values. We will discuss how with sufficient averaging in the coefficients of the polynomial (exponential in the size of the inputs), one can not only prove Bateman--Horn results on average but also pin down precise information about the distribution of prime values at finite but growing scales. We will prove that 100% of polynomials satisfy the appropriate analogue of the Poisson Tail Conjecture, in the sense that the distribution of the gaps between consecutive prime values around the average spacing is Poisson.

We will also study the frequencies of sign patterns of the Liouville function evaluated at the consecutive outputs of f; viewing f as a random variable, we establish the limiting distribution for every sign pattern. 
A key input behind all of our arguments is Leng's recent quantitative work on the higher-order Fourier uniformity of the von Mangoldt and M\"obius functions (in turn relying on Leng, Sah, and Sawhney's quantitative inverse theorem for the Gowers norms).

This talk is based on joint work with Noah Kravitz and Max Xu. 

Solutions to many problems in number theory can be described using the theory of algebraic stacks. In this talk, I will describe a Diophantine equation, the so-called “generalized Fermat equation”, whose integer solutions correspond to points on an appropriate stacky curve: a curve with extra automorphisms at prescribed points. Using étale descent over such a curve, we characterize local and global solutions to a family of such equations and give asymptotics for the local-global principle in the corresponding family of stacky curves. This is joint work with Juanita Duque-Rosero, Chris Keyes, Manami Roy, Soumya Sankar and Yidi Wang.