Special Year 2025-2026: Arithmetic Geometry, Hodge Theory, and O-minimality

Abstract: A result of Simpson characterizes subspaces that are (irreducible) algebraic varieties on both sides of the rank one Riemann-Hilbert correspondence. This result suggests an underlying Ax-Schanuel statement. In joint work with Jacob...

Sponsored by Dr. John P. Hempel 

Organizer: Jacob Tsimerman

The workshop focused on recent developments in Hodge theory, which has emerged as both a unifying framework and powerful tool for problems in arithmetic and unlikely intersections. The goal...

I will discuss questions pertaining to geometric unlikely intersections and transcendence in the setting of torii in positive characteristic. This is based on work in progress joint with Anup Dixit, Philip Engel, and Ruofan Jiang. 

Every o-minimal structure determines a collection of "tame" or "definable" subsets of $bbR^n$. We can then ask about the fragment of complex geometry present in the structure: Which holomorphic functions are definable, and which spaces are cut out...

In this talk I will introduce a bi-\bar{Q}-structure on Shimura varieties, propose a hyperbolic analytic subspace conjecture (analogue of Wüstholz’s analytic subgroup theorem in this context), and explain its consequence on quadratic relations...

In 2011 Bourgain and Rudnick showed that if $\gamma$ is a curve of non-vanishing curvature on the 2d standard flat torus, then there are no Laplace eigenfunctions of arbitrarily large eigenvalues containing $\gamma$ in their nodal set. We show that...

I'll talk about the o-minimal structures R_LN and R_{LN,exp} where one has an effective form of the finiteness property of o-minimality. Unlike the more classical structure of Pfaffian function, R_{LN,exp} contains the period integrals for aribtrary...

Since the work of Jacobi and Siegel, it is well known that Theta series of quadratic lattices produce modular forms. In a vast generalization, Kudla and Millson have proved that the generating series of special cycles in orthogonal and unitary...

Since the work of Jacobi and Siegel, it is well known that Theta series of quadratic lattices produce modular forms. In a vast generalization, Kudla and Millson have proved that the generating series of special cycles in orthogonal and unitary...

Many geometric spaces carry natural collections of special submanifolds that encode their internal symmetries. Examples include abelian varieties and their sub-abelian varieties, locally symmetric spaces with their totally geodesic subspaces, period...