Aaron Bertram

Title: Stability of vector bundles on surfaces is inherently complex

Abstract: The slope of a vector bundle on a curve is unambiguously defined as the ratio of the degree to the rank. This allows for an unambiguous definition of (families of) stable vector bundles, which have quasi-projective coarse moduli. The surface case is really quite different. Gieseker (or Mumford) slopes only tell part of the story, and there is a rich geometry of projective coarse moduli spaces of stable complexes of vector bundles on a surface, indexed by cohomology classes of the surface. The role of positivity is crucial, and the key result that I will talk about is recent work of Bayer and Macri on the positivity of the determinant line bundle on moduli.

Ana-Maria Castravet 

Title: Birational geometry of moduli spaces of stable n-pointed rational curves

Abstract: TBA

Sabin Cautis

Title: Categorical Heisenberg actions on Hilbert schemes of points

Abstract: We define actions of certain Heisenberg algebras on the Hilbert schemes of points on ALE surfaces. This lifts constructions of Nakajima and Grojnowski from cohomology to K-theory and derived categories of coherent sheaves. This action can be used to define Lie algebra actions (using categorical vertex operators) and subsequently braid group actions and knot invariants.

Zhiyuan Li

Title: Noether-Lefschetz conjecture and special cycles on Shimura varieties

Abstract: The Noether-Lefscehtz divisors on moduli space of quasi-polarized K3 surfaces parameterize the K3 surfaces with Picard number greater than one. Maulik and Pandharipande have conjectured that they span the Picard group with rational coefficients.It is well-known that this moduli space is a Shimura variety of orthogonal type. More generally, as I will explain, such question is equivalent to ask wether the special cycles of a Shimura variety can span low degree cohomology groups. In this talk, I will first talk about some geometric results of this conjecture and then its relation to automorphic representation theory. The recent progress and a sketch of proof of the most general case will be discussed in this direction.

Aaron Pixton

Title: The tautological ring of the moduli space of stable curves

Abstract: The tautological ring of the moduli space of smooth curves of genus g is the subring of its Chow ring generated by the kappa classes. This ring was introduced by Mumford in the 1980s and conjectural descriptions of its structure were given by Faber and Faber-Zagier in the 1990s. When the moduli space of smooth curves is compactified to form the moduli space of stable curves, the tautological ring gains additional combinatorial structure and becomes significantly larger. I will discuss some recent developments in the study of this larger tautological ring that were motivated by the conjectures of Faber and Faber-Zagier.

Aleksey Zinger

Title: Qualitative properties of Gromov-Witten invariants

Abstract: 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 the appropriate number of 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.