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 quasiprojective 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. 


Title: Birational geometry of moduli spaces of stable npointed 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 Ktheory 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: NoetherLefschetz conjecture and special cycles on Shimura varieties Abstract: The NoetherLefscehtz divisors on moduli space of quasipolarized 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 wellknown 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 FaberZagier 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 FaberZagier. 
Aleksey Zinger 
Title: Qualitative properties of GromovWitten 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) GromovWitten invariants. Physical considerations suggest that primary GWinvariants of CalabiYau threefolds, of any given genus, grow at most exponentially in the degree. For the genus 0 and 1 GWinvariants 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 GWinvariants of P^3. I will describe a formula that presents generating functions for the genus 0 GWinvariants of any complete intersection with any number of marked points as linear combinations of derivatives of a wellknown generating function for GWinvariants with 1 marked point. Estimates on the coefficients lead to bounds on GWinvariants 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. 