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

Abstract: 
I'll start by discussing some work with Binyamini, Schmidt, and Thomas in which we prove a uniform Manin-Mumford result for products of CM elliptic curves. I'll show how we apply this to obtain an effective Andre-Oort result for fibre...

Abstract: We study a non-commutative counterpart $C_n(X)$ of the symmetric product $Sym^n X$ of a space $X$, defined as the space of nxn-matrix valued points on $X$. Our main result will be a precise formula for the cohomology of this space in great...

Abstract: In 2019, Dimitrov proved the Schinzel-Zassenhaus Conjecture. Harry Schmidt and I extended his general strategy to cover some dynamical variants of this conjecture. One common tool in both results is Dubinin's Theorem on the transfinite...

Abstract: Since Langlands's earliest paper on his now famous program, canonical integral models of Shimura varieties have occupied a central role in modern number theory. Steady progress has been made in the intervening 50 years toward the correct...

Abstract: In this talk, we construct a Zariski open dense locus in $M_g$ on which the Beilinson-Bloch height of the Gross-Schoen and Ceresa cycles is a Weil height, i.e. it has a lower bound and satisfies the Northcott property. This implies a...

Abstract:  Applications of o-minimality to unlikely intersection problems usually begin with the observation that the relevant analytic covering maps are definable.  However, this observation is almost never literally true in that the maps are...

Abstract: Which complex analytic spaces can arise as the universal cover of a complex algebraic variety? Motivated by this question, Shafarevich asked whether the universal cover of a smooth projective variety X is always holomorphically convex —...

Abstract: Given an algebraic variety over a number field F, one can attach to it its etale cohomology groups, etale fundamental group, and higher etale homotopy groups, all equipped with an action of the absolute Galois group of F. The Galois action...

Abstract: We will discuss a proof that the integral Hodge conjecture is false for a very general abelian variety of dimension ≥ 4. Associated to any regular matroid is a degeneration of principally polarized abelian varieties. We introduce a new...

Abstract: I will discuss the general strategy of studying arithmetic objects, such as p-adic Galois representations, L-values, and automorphic forms, as members of (p-adic) families and explain how this can make certain questions more accessible. In...