Georgia Institute of Technology College of Sciences

Arithmetic Statistics is the study of arithmetic objects as they vary in a family. This is the framework used to analyze distributional properties of discriminants and class groups of number fields, ranks and Selmer groups of elliptic and hyperelliptic curves, low-lying zeroes and central values of L-functions, and many other key objects and invariants. An interrelated web of conjectures (Malle, Cohen--Lenstra, Poonen--Rains, Goldfeld, Katz--Sarnak, Loughran--Santens, and their many refinements) give quite precise conjectural descriptions (based on heuristical models) of the leading terms and asymptotic constants of the counts arising in these distributional questions. However, the secondary terms of the counts are much more mysterious, and very little is known (or even conjectured) about them.

In this talk, I will discuss some of the main distributional questions in arithmetic statistics, give some theoretical and practical explanations of why these secondary terms are so important, and end with explaining a series of results, joint with Manjul Bhargava, Takashi Taniguchi, and Jacob Tsimerman, in which some of these secondary terms are proved to exist, and computed.

Fractal sets arise naturally when classical notions of length, area, and smoothness are no longer adequate. A central theme in geometric measure theory is to understand how the dimension and geometry of a fractal set are reflected in its projections. Even some of the simplest questions of this type lead to surprisingly difficult problems connecting geometric measure theory with harmonic analysis, combinatorics, and number theory.

One classical example is the Favard length problem, also known as Buffon's needle problem. Given a planar set, its Favard length is the average length of its orthogonal projections. A theorem of Besicovitch implies that a purely unrectifiable set of finite length has Favard length zero. For self-similar sets obtained as limits of finite Cantor constructions, however, a much more quantitative question remains: how quickly does the Favard length of the finite approximations tend to zero? Despite the simple formulation, determining the correct decay rate remains a major open problem.

I will describe the geometry behind the classical problem, some of the main ideas that have led to progress, and what happens when linear projections are replaced by nonlinear families. This leads naturally to a nonlinear analogue of Buffon's needle problem—a Buffon circle problem—and to new questions about unions of circles, nonlinear projections, and the geometry of fractal sets. I will discuss recent results in this direction and the new phenomena that appear when lines are replaced by curves.

Working productively with modern AI tools requires looking beyond the products as they are currently packaged, identifying the underlying capabilities that are useful for our own goals, and then building systems to make those capabilities an honest net benefit in practice. The aim is to make the tools suit our interests while preserving what we value in mathematics. I will discuss this perspective through examples centered on research and verification.

Coupled oscillators appear in a large number of applications: e.g. in biological, chemical sciences, neuro science, power grids, and many more fields. They appear in nature: fireflies flashing in sync with each other is one fun situation.

In 1974, Yoshiki Kuramoto proposed a simple, yet surprisingly effective model for oscillators. We consider homogeneous Kuramoto systems (we will define these notions!). They are determined from a finite graph. In this talk, we describe some of what is known about long term behavior of such systems (do the oscillators self-synchronize? or are there other, "exotic" solutions?), and then relate these systems to systems of polynomial equations. We use algebra, computations in algebraic geometry, and algebraic geometry to study equilibrium solutions to these systems. We will see how computations using algebraic geometry and my computer algebra system Macaulay2 finds all graphs with at most 8 vertices (i.e. 8 oscillators) which have exotic solutions.

Note: we assume essentially NO dynamical systems nor algebraic geometry in this talk! This talk should be understandable to a general mathematical audience. The parts of the talk that are new represent joint work with Heather Harrington and Hal Schenck, and also Steve Strogatz and Alex Townsend.