Video Lectures

Given a family X→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 map S→D/Γ. We study the...

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 powers of the...

We study a non-commutative counterpart Cn(X) of the symmetric product SymnX 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 generality, but we...

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 diameter of...

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 formulation...

In this talk, we construct a Zariski open dense locus in Mg

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 generic positivity...

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 definable on a...

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 — that is...

Why is it so hard to prove P != NP, or even to prove super-linear circuit lower bounds? While we often blame a lack of combinatorial ingenuity, the bottleneck might be more fundamental: the logical strength of our mathematical tools.

This series of...

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 on etale...