Abstracts

Jarod Alper: Slice theorems for stacks and applications

We will begin by discussing the following theorem proven in joint work with Jack Hall and David Rydh: every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is etale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group. After briefly discussing extensions of this result to arbitrary fields and base schemes, we will focus on applications. First, we will show how this result allows us to extend classical theorems concerning algebraic groups. Second, we will apply this theorem to construct projective moduli spaces of objects (such as semistable vector bundles over a smooth projective curve) which may have infinite automorphism groups.

Michel Brion: Commutative algebraic groups up to isogeny.

The objects of the talk are the commutative group schemes of finite type over a field. They form an abelian category, which is generally quite complicated: it does not have enough projectives or injectives, its homological dimension can be arbitrarily large... We will discuss the category of commutative algebraic groups up to isogeny, which turns out to be much simpler. In particular, it has homological dimension 1, and is equivalent to the category of all left modules of finite length over a certain non-commutative ring (explicit but huge).

Emanuele Macri: Derived categories of cubic fourfolds and non-commutative K3 surfaces

The derived category of coherent sheaves on a cubic fourfold has a subcategory which can be thought as the derived category of a non-commutative K3 surface. This subcategory was studied recently in the work of Kuznetsov and Addington-Thomas, among others. In this talk, I will present joint work in progress with Bayer, Lahoz, Stellari and with Lahoz, Nuer, Perry, on how to construct Bridgeland stability conditions on this subcategory. This proves a conjecture by Huybrechts, and it allows to start developing the moduli theory of semistable objects in these categories, in an analogue way as for the classical Mukai theory for (commutative) K3 surfaces. I will also discuss a few applications of these results.

Dmitri Orlov: Geometric realizations of noncommutative varieties and phantoms

In my talk I am going to discuss such phenomena as phantom and quasi-phantom categories which appear as admissible subcategories in derived categories of coherent sheaves on fake del Pezzo surfaces. They represent examples of smooth and proper noncommutative varieties whose additive invariants are almost or completely trivial. I am also going to discuss general notions of noncommutative varieties and their geometric realizations.

Burt Totaro: Hodge theory of classifying spaces

The goal of this talk is to create a correspondence between the representation theory of algebraic groups and the topology of Lie groups. The idea is to study the Hodge theory of the classifying space of a reductive group over a field of characteristic p, the case of characteristic 0 being well known. The approach yields new calculations in representation theory, motivated by topology.

Giulia Saccà: Degenerations of hyperkähler manifolds

The problem of understanding semistable degenerations of K3 surfaces has been greatly studied and is completely understood. The aim of this

talk is to present joint work in progress with J. Kollár, R. Laza, and C. Voisin generalizing some of these results to higher dimensional hyperkähler (HK) manifolds. I will also present some applications, including a generalization of theorem of Huybrechts to possibly singular symplectic varieties and shortcuts to showing that certain HK manifolds are of a given deformation type.

Justin Sawon: On the topology of compact holomorphic symplectic manifolds

In this talk we will describe some results on the Betti, Hodge, and Chern numbers of compact holomorphic symplectic manifolds. In (complex) dimension four one can universally bound all of these invariants (Beauville, Guan). In higher dimensions it is still possible to find some bounds, in particular for the second Betti number. We will describe how such bounds are related to the fundamental question: are there finitely many deformation classes of compact holomorphic symplectic manifolds in each dimension?

Jacob Tsimerman: Ax-Schanuel for Shimura Varieties

(joint with N.Mok, J.Pila) Shimura varieties (S​) are uniformized by symmetric spaces (H), and the uniformization map π: H → S is quite transcendental. Understanding the interaction of this map with algebraic structures is of particular interest in arithmetic, as it is a necessary ingredient for the modern approaches to the Andre-Oort and Zilber-Pink conjectures. We establish an analogue of the Ax-Schanuel theorem in this context, which essentially says that any atypical algebraic relations between subvarieties in H and S are governed by Shimura subvarieties.

Mini-Talks

Omprokash Das: On the abundance problem for threefolds in characteristic p>5

Brian Hwang: Modeling arithmetic moduli spaces of abelian varieties by degenerating Grassmannians

Kuan-Wen Lai: Cremona transformations and derived equivalences of K3 surfaces

Patrick McFaddin: Zero-cycles with coefficients for some twisted homogeneous varieties

Lei Song: Direct images of twisted dualizing sheaves

David Stapleton: The degree of irrationality of hypersurfaces in some rational homogeneous spaces

Martin Ulirsch: The 2-category of toric stacks

Yuan Wang: Rational curves and generic ampleness of tangent bundles

Ruijie Yang: Higher syzygies of surfaces via the geometry of nested Hilbert schemes

Zheng Zhang: A generic global Torelli theorem for certain Horiwaka surfaces