Georgia Institute of Technology College of Sciences

Recent work studying relations between moduli spaces in algebraic and tropical geometry reveals new and previously unexpected structures in the weight zero cohomology of moduli of curves and abelian varieties. Surprisingly, the motivic Lie algebra, a free Lie algebra with one generator in each odd degree greater than or equal to 3, whose representations control the category of mixed Tate motives over Z, appears in the weight zero cohomology of both of these spaces, with no natural map between the two. I will discuss these structures and highlight recent work of Amy Bradford that connects these two copies of the motivic Lie algebra, through the weight zero cohomology of the moduli of curves of compact type.

In this talk I will present a complete set of Legendrian Kirby moves relating diagrams of the same contact 3–manifold, yielding a contact analogue of Kirby's theorem. Our approach refines Avdek's ribbon moves by using Gervais' presentation of mapping class groups using only nonseparating curves, together with explicit constructions of Legendrian representatives. This is a joint work with with Marc Kegel and Eric Stenhede.

Second of two talks.

In Bernoulli percolation, we place edges independently with probability p between vertices of a lattice, and study the resulting random geometry. In the first talk, I will give a friendly introduction to the model and discuss some of the motivating questions and the history of methods in the high dimensional setting. In the second talk, I will discuss our recently developed tool, the incipient infinite cluster, and describe how we use it to derive the exact limiting distribution of distances within clusters, as well as prove sharp asymptotics for k-point connectivity functions. These results together establish several aspects of the conjecture that the scaling limit of high dimensional critical clusters is the Integrated Super-Brownian Excursion. Based on joint work with Shirshendu Chatterjee, Jack Hanson, and Philippe Sosoe.

Weil's converse theorem characterizes modular forms by the functional equations of their L-functions and all of their twists by Dirichlet characters. Farmer, Koutsoliotas and Lemurell conjectured that the twist assumption is redundant, that is, a degree 2 Euler product whose L-function has a suitable functional equation should come from a Hecke eigenform. We report on some progress towards this conjecture based on work with Andrew Booker and Min Lee, as well as some recent LLM assistance.

The inverse Galois problem, a foundational question in number theory, asks whether every finite group $G$ can be realized as the Galois group of a field extension of the rational numbers. Malle's conjecture is a refined version of the inverse Galois problem which predicts the asymptotic number of such extensions.  In joint work with Ishan Levy, we prove a version of Malle's conjecture, computing the asymptotic growth of the number of Galois $G$ extensions of $\mathbb F_q(t)$, for $q$ sufficiently large and relatively prime to $|G|$. The key input is a computation of the stable cohomology of certain Hurwitz spaces.

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.

The dimer model, i.e. random perfect matchings of a bipartite graph, is a classical object about which much is known. As soon as one biases the probability measure by edge weights which are themselves random, very little is known rigorously, though physicists have studied such models for several decades and made extensive predictions. I will discuss a new integrable model in this class (the Gamma-disordered Aztec diamond) which allows us to prove results on the free energy, and also exhibits surprising relations to integrable polymer models which lead to probabilistic results on tilings. Joint work with Maurice Duits (KTH), https://arxiv.org/abs/2512.03033.