Photo by Brian Wolfe / CC BY NC
| Registration | Program | Abstracts |
Photo by Brian Wolfe / CC BY NC |
University of Michigan | Ann Arbor |
In an effort to better understand the different constructions and computations of Witten's class we are inviting several experts to speak on the subject.
Partial financial support available for graduate students and postdocs. Please contact Felix Janda for information.
| Saturday, January 21, 2017 | ||
|---|---|---|
| 9:30-10:00 | Coffee and Bagels | |
| 10:00-11:15 | Alexander Polishchuk | FJRW-theory via matrix factorizations. I |
| 11:30-12:30 | Alessandro Chiodo | Hodge integrals and the Witten top Chern Class |
| 14:00-15:00 | Dimitri Zvonkine | Teleman's classification of semisimple CohFTs - I |
| 15:15-16:15 | Aaron Pixton | Explicit formulas for Witten's r-spin class |
| 16:15-16:45 | Coffee break | |
| 16:45-17:45 | Alexander Polishchuk | FJRW-theory via matrix factorizations. II |
| Sunday, January 22, 2017 | ||
| 8:30-9:45am | Qile Chen | A virtual cycle construction for p/q-spin structures |
| 9:45-10:15 | Coffee and Bagels | |
| 10:15-11:15 | Dimitri Zvonkine | Teleman's classification of semisimple CohFTs - II |
| 11:30-12:30 | Aaron Pixton | Quasimodularity of elliptic curve invariants via the double ramification cycle |
| 13:30-14:30 | Felix Janda | Conjectural formula for Witten's r-spin class |
All talks will take place in East Hall 4096.
Buryak, Dubrovin, Guéré and Rossi have recently found a new approach to integrable hierarchies involving r-spin curves and the Hodge classes. The standard virtual Witten top Chern class is well-defined but difficult to compute beyond genus zero. I will try to explain why the integral against the Hodge class can be systematically solved via Grothendieck-Riemann-Roch in all genera.
While the work of Pandharipande, Pixton and Zvonkine gives an explicit formula for the push-forward of Witten's class to the moduli of stable curves, we still cannot compute Witten's class on the moduli space of r-spin curves. In my talk, I want to describe a conjectural formula for Witten's class in terms of loci of effective spin structures.
I will explain how to obtain explicit formulas for Witten's r-spin class using Teleman's classification of semisimple cohomological field theories. I will also discuss how these formulas yield tautological relations. This is joint work with Rahul Pandharipande and Dimitri Zvonkine.
Double ramification cycles parametrize curves that admit maps to the projective line with specified ramification profiles over zero and infinity. I will briefly discuss a formula for these cycles in terms of tautological classes, and then I will explain how to apply this formula to answer questions about the quasimodularity of Gromov-Witten invariants of an elliptic curve.
I will describe the construction (joint with Vaintrob) of the Witten’s virtual class on the moduli spaces of higher-spin curves, associated with a nondegenerate quasihomogeneous polynomial. The key features of the construction is the use of categories of matrix factorizations and of Hochschild homology.
Following Teleman's paper we introduce three types of cohomological field theories (CohFTs): open CohFTs on M_{g,n}, open CohFTs with fixed boundaries on certain torus bundles over M_{g,n}, and the Kontsevich-Manin CohFTs on Mbar_{g,n}. Simpler CohFTs in this list are obtained by restriction or by pull-back from the more complicated ones. We will prove the classification theorem for each type, starting by the simpler ones.