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

A category of motives is an axiomatic framework in which several cohomology theories from algebraic geometry are represented. While Morel and Voevodsky's classical framework of motivic homotopy theory focused on $A^1$-invariant cohomology theories...

I will discuss how the Lang–Trotter conjecture for pairs of elliptic curves implies new cases of the Zilber–Pink conjecture for curves in $\mathcal{A}_3$. Unlike previous results for curves in $\mathcal{A}_g$, our result does not rely on any...

Around 1921 Polya asked about the algebraicity of the indefinite integral of an algebraic function, in terms of the integrality of the coefficients in a power-series expansion. Polya nicely answered the special case of rational functions, using...

The Pila–Zannier strategy has proved to be a particularly fruitful approach: first applied to reprove the Manin–Mumford conjecture, it was later exploited decisively in proofs of the André–Oort conjecture. Crucially relying on a recent counting...

Ideas from tame geometry have recently begun to find their way into quantum field theory and string theory, suggesting that consistent effective theories and their observables may admit definable descriptions of finite complexity. In this talk I...

Abstract: Joint project with Pavao Mardesic, Laura Ortiz-Bobadilla, and Jessie Pontigo-Herrera.

Hibert's 16th problem asks for an upper bound on the number of limit cycles of planar polynomial vector fields. For polynomial perturbations $\dH+\epsilon...

Abstract: I will talk about a joint work with Antoine Sedillot. We study sup-norms of sections of metrized line bundles for families of arithmetic varieties. Using Riemann-Zariski spaces we obtain formulas that imply new definability results in the...

Abstract: Constructing completions of period mappings with significant geometric and Hodge-theoretical meaning is an important topic in Hodge theory and its applications. There are rich theories for the classical case in which the period domain is...

Abstract: The anabelian phenomenon may be viewed as an arithmetic analogue of Mostow rigidity: it predicts that certain varieties can be reconstructed from their arithmetic fundamental groups. A celebrated result of S. Mochizuki shows that...

Abstract: Given a family $X \rightarrow S$, one may consider the corresponding fiber-wise (quasi-) period integrals as (multi)-functions on S. Built out of these using a flag variety, one obtains variation of (mixed) hodge structures giving period...