For smooth projective varieties over Q, is the existence of a rational point preserved under derived equivalence? First I'll discuss why this question is interesting, and what is known. Then I'll show that the answer is no, giving two counterexamples: an abelian variety and a torsor over it, and a pair of moduli spaces of sheaves on a K3 surface.

I will discuss the asymptotic non-vanishing of syzygies for products of projective spaces, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.

The purpose of this talk is to encourage people to think about purity questions. We will discuss a strong form of purity for morphisms of relative dimension 0. For morphisms of relative dimension 1 we relate the purity question to one on purity for families of curves.

Albanese maps and fundamental groups of varieties with many rational points over function fields

In this talk we will discuss topological properties of varieties with many rational points over a function field, and present joint work-in-progress with Erwan Rousseau. More precisely, we define a smooth projective variety X over the complex numbers to be geometrically-special if there is a dense set of closed points S in X such that, for every x in S, there is a pointed curve (C,c) and a sequence of morphisms (C,c)->(X,x) which covers C x X, i.e., the union of their graphs is Zariski-dense in C x X. Roughly speaking, a variety is geometrically-special if it satisfies density of "pointed" rational points over some function field. Inspired by conjectures of Campana on special varieties and Lang on hyperbolic varieties, we prove that every linear quotient of the fundamental group pi_1(X) of such a variety is virtually abelian.

Using the theory of polarised flag type quotients, we determine mass formulae for all principally polarised supersingular abelian threefolds defined over an algebraically closed field k of characteristic p. We combine these results with computations of the automorphism groups to study Oort's conjecture; we prove that every generic principally polarised supersingular abelian threefold over k of characteristic >2 has automorphism group \( \mathbb{Z}/2\mathbb{Z} \).

A scheme X is a topological space---which we denote by |X|---and a sheaf of rings on the open subsets of |X|. We study the following natural but seldom considered questions. How to read off properties of X from |X|? Does |X| alone determine X? Joint work with Max Lieblich, Martin Olsson, and Will Sawin.

I will review some classical questions on the relationship between nonnegative polynomials and sums of squares in R^n and briefly survey their generalizations to the context of real projective varieties. The results presented in this talk are joint work with Greg Blekherman, Rainer Sinn and Greg G. Smith.

The generic Torelli theorem for hypersurfaces of degree d and dimension n-1 was proved by Donagi in the 90's. It works under the assumption that d is at least 7 and d does not divide n+1, which in particular excludes the Calabi-Yau case in all dimensions. We prove that the generic Torelli theorem for hypersurfaces holds with finitely many exceptions. A key tool is the notion of Schiffer variation of a hypersurface and how to characterize them by looking at the variation of Hodge structure along them.