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de 11 avril 2016 à 15 juillet 2016
Institut Henri Poincaré
Europe/Paris timezone

Dimitri Zvonkine, Paris 6

Cohomological relations on the moduli space of stable curves

We construct a family of relations between tautological cohomology classes on the moduli space Mbar_{g,n} of stable curves. This family contains all relations known to this day and is expected to be complete and optimal. The construction uses the Frobenius manifold of the A_2 singularity, the 3-spin Witten class and the Givental-Teleman classification of semi-simple cohomological field theories (CohFTs). The plan of the three talks will be as follows.

1. An introduction to moduli space and its tautological cohomology ring; simplest examples of tautological relations.

2. Cohomological field theories and Witten's r-spin class. Witten's r-spin class is actually a family of cohomology classes on the space of stable maps, defined using the space of tensor r-th roots of the canonical line bundle. I will explain why these classes satisfy the axioms of a cohomological field theory (CohFT).

3. The Givental-Teleman classification of semi-simple CohFTs. I will explain the classification theorem, show how it applies to Witten's class and how one can deduce tautological relations from it. In the end I will compute several cohomological relations using our method.

This is a joint work with R. Pandharipande and A. Pixton.