Abstract: I will discuss new invariants of actions of finite groups on algebraic varieties and their applications (joint with A. Kresch, I. Cheltsov and Zh. Zhang).

Abstract: A theory of heights of rational points on stacks was recently introduced by Ellenberg, Satriano and Zureick-Brown as a tool to unify and generalize various results and conjectures about counting problems over global fields. In this talk I will present a moduli theoretic approach to heights on stacks over function fields inspired by twisted stable maps of Abramovich and Vistoli. For some well-behaved class of stacks, we obtain moduli spaces of points of fixed height whose geometry controls the number of rational points on the stack. I will outline an approach for more general stacks which is closely related to the geometry of the moduli space of vector bundles on a curve. This is based on joint work with Park and Satriano.

Abstract: I will explain a numerical invariant of complex varieties born out of the difference between algebraic and topological K-theory. In some cases, it is a birational invariant which is weaker than some known ones coming from unramified cohomology. However, it is also a derived invariant which lends itself to some calculation that reveals some surprising connections between the derived category of a variety, algebraic cycles and torsion in cohomology. As an application, we prove new cases of integral Hodge conjecture for some Fano varieties. This is all joint work with Nick Addington.

Abstract: Over 30 years ago, the work of Beauville and Deninger-Murre endowed the cohomology of an abelian scheme a (motivic) decomposition which splits the Leray filtration. This structure, now known as the Beauville decomposition, is induced by algebraic cycles obtained from the Fourier-Mukai coherent duality. In recent years, the study of Hitchin systems (e.g. the P=W conjecture) and compactified Jacobians suggests that there should exist an extension of the theory of Beauville decompositions for certain abelian fibrations with singular fibers, where the Leray filtration should be replaced by the perverse filtration. I will discuss some recent progress in this direction. In particular, I will present results in both the positive and the negative directions, where Lagrangian fibrations associated with hyper-Kähler manifolds and the tautological relations over the moduli of stable curves play crucial roles. Based on joint work with Younghan Bae, Davesh Maulik, and Qizheng Yin.

Abstract: A Lagrangian fibration of a projective hyper-Kähler manifold is called isotrivial if its smooth (abelian variety) fibers are all isomorphic to each other. Given an isotrivial fibration of a HK manifold f : X -> B with at least one rational section, we prove the following four descriptions of the fibration: (1) The common abelian variety fiber F is isogenous to the power of an elliptic curve E. (2) There are two types (called type A and B) of isotrivial fibrations with different behaviors. (3) If the elliptic curve factor E of the fiber has no CM by Q(\sqrt{-1}) or Q(\sqrt{-3}), then the fibration is necessarily of type A. (4) Every type A isotrivial fibration is birational to one of the two explicit constructions of isotrivial HK, starting from an elliptic K3 surface or an abelian surface. Our proof of the last statement (4) depends on the regularity conjecture on the base of a Lagrangian fibration. This is joint work with Radu Laza and Olivier Martin.

Abstract: A metric on a category assigns lengths to morphisms, with the triangle inequality holding. This notion goes back to a 1974 article by Lawvere. We'll start with a quick review of some basic constructions, like forming the Cauchy completion of a category with respect to a metric.

And then will begin a string of surprising new results. It turns out that, in a triangulated category with a metric, there is a reasonable notion of Fourier series, and an approximable triangulated category can be thought of as a category where many objects are the limits of their Fourier expansions. And some other ideas, mimicking constructions in real analysis, turn out to also be powerful. 

And then come two types of theorems: (1) theorems providing examples, meaning showing that some category you might naturally want to look at is approximable, and (2) general structure theorems about approximable triangulated categories. 

And what makes it all interesting is (3) applications. These turn out to include the proof of an old conjecture of Bondal and Van den Bergh about strong generation, a representability theorem that leads to a short, sweet proof of Serre's GAGA theorem, a proof of a conjecture by Antieau, Gepner and Heller about the non-existence of bounded t-structures on the category of perfect complexes over a singular scheme, as well as (most recently) a vast generalization and major improvement on an old theorem of Rickard's.

Abstract: We will investigate if some well known properties of log canonical singularities over the complex numbers still hold true over perfect fields of positive characteristic and over excellent rings with perfect residue fields. We will discuss both pathological behavior in characteristic p as well as some positive results for threefolds. We will see that the pathological behavior of these singularities in positive characteristic is closely linked to the failure of certain vanishing theorems in positive characteristic. Additionally, we will explore how these questions are related to the moduli theory of varieties of general type.

This is based on joint work with F. Bernasconi and Zs. Patakfalvi, as well as joint work with Q. Posva.

Abstract: I will explain a Hodge-theoretic proof for Hwang's theorem, which says that if the base of a Lagrangian fibration on an irreducible holomorphic symplectic manifold is smooth, then it must be projective space. The result is contained in a joint paper with Ben Bakker from last fall.

Abstract: Syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. Starting with the pioneering work of Mark Green on curves, numerous attempts have been made to extend these results to higher dimensions. Ein and Lazarsfeld proved that if A is a very ample line bundle, then K_X + mA satisfies property N_p for any m>=n+1+p. It has ever since been an open question if the same holds true for A ample and basepoint free. In recent joint work with Purnaprajna Bangere we give a positive answer to this question.