Registration | Program | Abstracts | University of Michigan | Ann Arbor |

In an effort to better understand the geometry of strata of abelian differentials, we are inviting several experts to speak on the subject.

- Dawei Chen (Boston College/IAS)
- Gavril Farkas (Humboldt/MSRI)
- Dmitry Korotkin (Concordia University)
- Adrien Sauvaget (Utrecht)
- Johannes Schmitt (ETHZ)
- Alex Wright (Michigan)

- Felix Janda
- Yongbin Ruan

- NSF-FRG grant
- University of Michigan math department

Partial financial support available for graduate students and postdocs. Please contact Felix Janda for information.

Saturday, February 23, 2019 | ||
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9:30-10:00 | Coffee & Tea | |

10:00-11:00 | Dawei Chen | Volume and intersection theory on strata of abelian differentials, I |

11:00-11:30 | Coffee & Tea | |

11:30-12:30 | Adrien Sauvaget | Volume and intersection theory on strata of abelian differentials, II |

14:30-15:30 | Gavril Farkas | Quadric rank loci on moduli of curves and K3 surfaces |

15:30-16:00 | Coffee & Tea | |

16:00-17:00 | Johannes Schmitt | Moduli spaces of twisted k-differentials and Pixton's cycle |

Sunday, February 24, 2019 | ||

9:00-9:30 | Coffee & Tea | |

9:30-10:30 | Dmitry Korotkin | Symplectic aspects of monodromy map on a Riemann surface and period coordinates |

10:30-11:00 | Coffee & Tea | |

11:00-12:00 | Alex Wright | GL(2,R) invariant subvarieties of the Hodge bundle |

All talks will take place in East Hall 4096.

Computing volumes of moduli spaces has significance in many fields. For instance, Wittenâ€™s conjecture regarding intersection numbers on the moduli space of curves has a fascinating connection to the Weil-Petersson volume. In this talk we will introduce a formula of intersection numbers on strata of abelian differentials that computes the Masur-Veech volume. If time permits, we will also talk about an application for computing saddle connection Siegel-Veech constants on flat surfaces.

Given two vector bundles E and F on a variety X and a morphism from Sym^2(E) to F, we compute the cohomology class of the locus in X where the kernel of this morphism contains a quadric of prescribed rank. Our formulas have many applications to moduli theory: (i) a simple proof of Borcherds' result that the Hodge class on the moduli space of polarized K3 surfaces of fixed genus is of Noether-Lefschetz type, (ii) an explicit canonical divisor on the Hurwitz space parametrizing degree k covers of the projective line from curves of genus 2k-1, (iii) a closed formula for the Petri divisor on the moduli space of curves consisting of canonical curves which lie on a rank 3 quadric and (iv) myriads of effective divisors of small slope on M_g. Joint work with Rimanyi.

The monodromy map for second order differential equation on a Riemann surface is a classical subject closely related to uniformization theory. Symplectic aspects of this map attracted a lot of attention recently both from mathematics side (S.Kawai etc) and physics side (works on supersymmetric Yang-Mills equations, Hitchin's equations etc). The goal of this talk is to show that this monodromy map can be effectively studied using elementary machinery developed in the framework of isomonodromy deformations by A.Kokotov and the speaker. The main technical tool is the period, or homological, coordinates on spaces of quadratic differentials and corresponding variational formulas. This formalism can be used to prove that the natural monodromy map between the space of quadratic differentials and the monodromy character variety is a symplectomorphizm. A few new fact about corresponding generating functions (known as "Yang-Yang" functions in physics literature) can also be established. The talk is based on joint work with B.Bertola and C.Norton.

In this talk we will explain the proof of the volume/intersection formula on strata of abelian differentials. In particular, we will show that the intersection numbers and the volumes satisfy two recursions that have different looks but are equivalent. The intersection recursion relies on the boundary behavior of the strata, the volume recursion arises from asymptotics of Hurwitz numbers of torus covers, and their equivalence is a combinatorial outcome by summing over certain oriented graphs in two ways.

Inside the moduli space of smooth genus g curves with n markings, there are the loci of curves admitting a meromorphic k-differential with prescribed zeros and poles at the marked points. We explain a compactification of these loci proposed by Farkas-Pandharipande and describe its components and their dimensions. We present a conjectural relation between a weighted fundamental class of the compactification with a tautological cycle defined by Pixton.

We will survey some of the points of connection between algebraic geometry and Teichmuller dynamics, focussing on the problem of classifying GL(2,R) invariant subvarieties of the Hodge bundle. The talk will touch on joint works with subsets of Chen, Eskin, Filip, McMullen, Mirzakhani, and Mukamel.