Dawei Chen

Title: Geometry of moduli spaces of differentials

Abstract: Moduli spaces of differentials with a given type of zeros and poles on Riemann surfaces provide a stratification of the Hodge bundle, whose study has broad connections to flat geometry and billiard dynamics. In this talk we will introduce this topic from the perspective of an algebraic geometer, with a focus on recent results and open problems.

Jack Huizenga

Title: Properties of general sheaves on Hirzebruch surfaces

Abstract: Let X be a Hirzebruch surface. Moduli spaces of semistable sheaves on X with fixed numerical invariants are always irreducible by a theorem of Walter. Therefore it makes sense to ask about the properties of a general sheaf. We consider two main questions of this sort. First, the weak Brill-Noether problem seeks to compute the cohomology of a general sheaf, and in particular determine whether sheaves have the "expected" cohomology that one would naively guess from the sign of the Euler characteristic. Next, we use our solution to the weak Brill-Noether problem to determine when a general sheaf is globally generated. A key technical ingredient is to consider the notion of prioritary sheaves, which are a slight relaxation of the notion of semistable sheaves which still gives an irreducible stack. Our results extend analogous results on the projective plane by Gottsche-Hirschowitz and Bertram-Goller-Johnson to the case of Hirzerbruch surfaces. This is joint work with Izzet Coskun.

Hsian-Hua Tseng

Title: A tale of four theories

Abstract: Around a decade ago the following four (C^*)^2-equivariant theories are proven to be equivalent:

  • Gromov-Witten theory of P^1 \times C^2 relative to three fibers;
  • Donaldson-Thomas theory of P^1\times C^2 relative to three fibers;
  • Quantum cohomology of Hilbert schemes of points on C^2;
  • Quantum cohomology of symmetric product stacks of C^2.
In this talk we'll discuss these four equivalence. We'll also sketch some new development, namely higher genus extensions of these equivalences (joint work with R. Pandharipande).

Xiaolei Zhao

Title: Canonical points on K3 surfaces and hyper-Kahler varieties

Abstract: The Chow groups of algebraic cycles on algebraic varieties have many mysterious properties. For K3 surfaces, on the one hand, the Chow group of 0-cycles is known to be huge. On the other hand, the 0-cycles arising from intersections of divisors and the second Chern class of the tangent bundle all lie in a one dimensional subgroup. A conjecture of Beauville and Voisin gives a generalization of this property to hyper-Kähler varieties. In my talk, I will recall these beautiful stories, and explain a conjectural connection between the K3 surface case and the hyper-Kähler case. If time permits, I will also explain how to extend this connection to Fano varieties of lines on some cubic fourfolds. This is based on a joint work with Junliang Shen and Qizheng Yin.