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

Abstract: I'll talk about some classical problems around orthogonal polynomials and harmonic analysis that fall into the "unlikely intersection" paradigm: they can be reformulated as questions about counting integers in definable sets whose...

Abstract: Given a local system on a complex algebraic variety, what are the subvarieties on which the monodromy drops? The talk will discuss these monodromy special loci, a natural generalisation of (the positive period dimension components of) the...

Abstract: : Let $g$:$X \rightarrow S$ be a smooth proper family over algebraic varieties over a number field $K$. Consider the statement $W(g,p)$: the weight-monodromy conjectures hold for all fibres of g defined over a finite extension $L/Q_p$. We...

Abstract: To show that the Gamma function, restricted to the positive real half-axis, generates an o-minimal structure over the real field, we had to show (in collaboration with Lou van den Dries) that the expansion of the real field by all...

Abstract: The Zilber-Pink conjecture is a far reaching and widely open conjecture in the area of "unlikely intersections" generalizing many previous results in the area, such as the recently established André-Oort conjecture. Recently the ``G...

Abstract: The classical Rees construction (of common use in commutative algebra and Hodge theory) interpolates between filtrations, viewed as $G_m$-equivariant vector bundles on the affine line, and their associated gradings. Various non-abelian...

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...