The proetale topology. Bhargav Bhatt (IAS)

(joint work with Peter Scholze) The proetale topology is a Grothendieck topology that is closely related to the etale topology, yet better suited for certain "infinite" constructions, typically encountered in l-adic cohomology.

I will first explain the basic definitions, with ample motivation, and then discuss applications. In particular, we will see why locally constant sheaves in this topology yield a fundamental group that is rich enough to detect all l-adic local systems through its representation theory (which fails for the groups constructed in SGA on the simplest non-normal varieties, such as nodal curves).

Moduli spaces of odd theta characteristics. Gavril Farkas (Humboldt University of Berlin)

I will discuss joint work with Verra concerning a complete birational classification of the moduli space of odd spin curves of genus g. In particular, for g<12, we find explicit unirational parametrizations of the moduli space, by constructing new models of the spin moduli space mirroring Mukai's well-known work on the structure of canonical curves of genus at most 9.

Stable cohomology of moduli of abelian varieties and compactifications. Samuel Grushevsky (Stony Brook University)

The cohomology of A_g, the moduli space of principally polarized complex g-dimensional abelian varieties, is the same as the cohomology of the symplectic group with integer coefficients Sp(2g,Z). The results of Borel on group cohomology imply that the cohomology H^k(A_g) is independent of g for g>k - and determine this stable cohomology. We discuss the (non)existence of stable cohomology for various toroidal compactifications of A_g, and in particular compute the cohomology of the perfect cone compactification in low degrees. We will also highlight the parallels with the moduli space of curves.

Joint work with K. Hulek and O. Tommasi.

Nef cones of Hyperkähler manifolds. Emanuele Macrì (OSU)

In this talk we discuss a conjecture of Hassett and Tschinkel on the structure of the nef and Mori cones of a projective Hyperkähler manifold M of K3^[n]-type, i.e., M is deformation equivalent to the Hilbert scheme of points on a K3 surface. The main result is a complete description of these cones in terms of a certain lattice -- introduced by Markman -- naturally associated to M.

I will present joint work with Arend Bayer on how to prove the main theorem for moduli spaces of sheaves on a K3 surface, by using derived category techniques and Bridgeland stability. If time permits, I will also sketch how to extend the result to all Hyperkähler manifolds of K3^[n]-type, as recently proven by Bayer-Hassett-Tschinkel and Mongardi.

Kodaira dimension and zeros of holomorphic one-forms. Mihnea Popa (UIC)

I will explain joint work with C. Schnell in which we show that every holomorphic one-form on a smooth projective variety of general type must vanish at some point, together with an appropriate generalization to arbitrary Kodaira dimension. This answers positively a conjecture of Hacon-Kovács and Luo-Zhang. The proof uses generic vanishing theory for Hodge D-modules on abelian varieties.

Cohomology of moduli spaces of genus 2 curves and the Gorenstein conjecture. Orsola Tommasi (Leibniz Universität Hannover)

A main theme in the study of the cohomology of moduli spaces of curves is the study of the tautological ring, a subring generated by certain geometrically natural classes. An open question is whether the tautological ring is a Gorenstein ring, as conjectured by Carel Faber in the case of smooth curves without marked points. In this talk we discuss an approach that allows to detect the existence of non-tautological classes in the cohomology ring of the moduli space of stable curves of genus 2 with sufficiently many marked points, such as those constructed by Graber and Pandharipande for \bar M_{2,20}. We use this to prove that the Gorenstein conjecture does not hold for these spaces.

This is joint work with Dan Petersen (KTH, Stockholm).

Qualitative properties of Gromov-Witten invariants. Aleksey Zinger (Stony Brook University)

Over 15 years ago, di Francesco and Itzykson gave an estimate on the growth (as the degree increases) of the number of plane rational curves passing through 3d-1 general points. This provides an example of an upper bound on (primary) Gromov-Witten invariants. Physical considerations suggest that primary GW-invariants of Calabi-Yau threefolds, of any given genus, grow at most exponentially in the degree. For the genus 0 and 1 GW-invariants of projective complete intersections, this can be seen immediately from the known mirror formulas. Maulik and Pandharipande expect that such a bound in higher genera can be deduced from a suitable bound on the genus 0 descendant GW-invariants of P^3.

I will describe a formula that presents generating functions for the genus 0 GW-invariants of any complete intersection with any number of marked points as linear combinations of derivatives of a well-known generating function for GW-invariants with 1 marked point. Estimates on the coefficients lead to bounds on GW-invariants of all projective complete intersections. Even without any estimates, the structure of this formula leads to fascinating vanishing results, which do not appear to have any geometric explanation at this point.