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

Abstract: From the outset, topology has played an important role in the study of o-minimal structures. The central focus has been on developing the theory of o-minimality as a framework for 'tame topology', built upon the natural and well-behaved...

Abstract: Abstract: Let $C$ be a curve defined over a finite field, and let $X/C$ be a non-isotrivial family of K3 surfaces. In joint work with Maulik-Tang, under a compactness assumption (an assumption removed in later work by Tayou), we prove that...

Abstract: A classical theorem due to Borel states that every holomorphic map from a poly-punctured disk into a Shimura variety (with torsion-free level structure) extends holomorphically across the punctures to the minimal compactification. As a...

Abstract: I will talk about mod p versions of the Mumford—Tate and André—Oort conjectures. Via a notion of formal linearity, the two conjectures, together with a third one ($mod p $Ax—Lindemann), are closely entangled with each other — much closer...

Abstract: I will discuss work in progress with M. Orr (Manchester) and G. Papas (Weizmann) on the Zilber-Pink conjecture for $Y(1)^3$. This is known for so-called asymmetric curves by the 2012 work of Habegger-Pila. More recently, an approach known...

Abstract: We will survey various recent results around the distribution of the Hodge locus of a (mixed) variation of Hodge structures. Various concrete applications to moduli spaces will also be presented.

Abstract: We prove that an algebraic flat connection has $ {\mathbb R}_{\rm{ an, \  exp}}$-definable flat sections if and only if it is regular singular with unitary monodromy eigenvalues at infinity, refining previous work of Bakker–Mullane. This...

Abstract: I’ll describe recent work for approaching certain functional transcendence problems through a combination of model theory and differential Galois theory. This is based on joint work with Blazquez-Sanz, Casale, and Nagloo.

Abstract: Let $k$ be a field and $X$ a geometrically connected variety over $k$. The Tate or degeneracy locus of a $l$-adic local system on $X$ is the etale counterpart of the Hodge locus of a VHS. While in the last decade tremendous progresses have...

Abstract: The cohomology of a family of algebraic varieties carries a number of interrelated structures, of both Hodge-theoretic and arithmetic flavors. I’ll explain joint work with Josh Lam developing analogues of some of these structures and...