## The Eighth Advanced Seminar in Symplectic Geometry and Topological Field Theory

**Speaker(s): ** Prof. Motohico Mulase (UC. Davis), Prof. Emanuel Scheidegger (BICMR)

**Time: **October 8 - October 12, 2018

**Venue: ** Room 77201, Jingchunyuan 78, BICMR

**MinicourseⅠ**

**: Topological recursion in geometry (**

**Motohico Mulase)**

In these lectures, I will discuss topological recursion from a geometric perspective. Topics of talks will include the following:

1. Combinatorial origin of topological recursion.

Although topological recursion was discovered in random matrix theory by Eynard and Orantin in 2007 after preceding works by many others, the simplest mathematical origin is in combinatorial counting of graphs on surfaces, which is also closely related to lattice point counting on the moduli space of smooth pointed curves considered by Norbury. I will review these counting problems and introduce the concept of topological recursion.

2. Graphical expression of CohFT.

The CohFT formulated by Kontsevich-Manin relates Frobenius algebras and cohomology groups of moduli spaces of curves. The surprising result of Madsen-Weiss solving the Mumford Conjecture provides a powerful tool for classifying all semi-simple CohFT, which was carried out by Teleman. I will present a different perspective of CohFT, using the graphical ingredients appeared in Topic 1 above. In this talk, the relation between Frobenius algebras and the Hopf algebra in the work of Madsen-Weiss will be explained from the graphical point of view. This part will be based on my most recent joint work with Olivia Dumitrescu (in preparation).

3. Quantum curves in geometry.

We start with describing the key examples of quantum curves, then place these in the context of quantizing Hitchin spectral curves. The quantization is compared with the construction of opers, conjectured by Gaiotto. This particular conjecture of Gaiotto has been recently solved. I will give a brief survey on this result.

4. Topological recursion and Hitchin fibration.

Topological recursion has been transplanted in the context of Hitchin fibtaion by Dumitrescu and myself in 2014. Recently David Baraglia discovered a surprising relation between geometry of moduli spaces of Higgs bundles and topological recursion. I will give a survey of this new direction of research.

**MinicourseⅡ:Landau-Ginzburg Models and Matrix Factorizations**** (****Emanuel Scheidegger)**

Equivariant matrix factorizations are a special class of modules associated to the singularity given by $W$ and play the role of boundary conditions in the open/closed Landau-Ginzburg models. They are the analog of coherent sheaves on projective varieties and form a triangulated category. Using matrix factorizations, Polishchuk and Vaintrob gave an alternative way of constructing the Fan-Jarvis-Ruan-Witten theory.

We will conclude with some new results on central charge functions in the category of matrix factorizations. If time permits, relations to other concepts such as mirror symmetry will be exhibited.

If you are interested to join in the workshop, please send email to haoying@bicmr.pku.edu.cn with your name, affiliation, position and phone number. We will further contact you to confirm your registration and accommdation.

Contact:

Ms.Ying Hao

No.78 Jingchunyuan,Peking University, Beijing,China 100871

Tel: 010-62744130

E-mail: haoying@bicmr.pku.edu.cn