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

Time: 2018-10-08
Published By: Ying Hao

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

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

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

Organizer:

Bohan Fang (Peking University)

Xiaobo Liu (Peking University)

Speaker:

Motohico Mulase (UC Davis)

Emanuel Scheidegger (Peking University)

Chenchang Zhu (Georg-August-Universität Göttingen)

Schedule:

09:30-11:30 Emanuel Scheidegger

11:30-14:00 Lunch Break

14:00-16:00 Motohico Mulase

09:30-11:30 Motohico Mulase

11:30-14:00 Lunch Break

14:00-16:00 Emanuel Scheidegger

09:30-11:30 Motohico Mulase

11:30-14:00 Lunch Break

14:00-16:00 Free

09:30-11:30 Emanuel Scheidegger

11:30-14:00 Lunch Break

14:00-16:00 Chenchang Zhu

09:30-11:30 Motohico Mulase

11:30-13:00 Lunch Break

13:00-15:00 Emanuel Scheidegger

Title & Abstract:

Title:

Abstract: 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.The canonical extensions of automorphic vector bundles on Shimura varieties are vector bundles whose global sections characterize holomorphic automorphic forms, and in this series of talks we show that the Hecke module structures on their higher cohomology also come from automotphic forms. As people have associate Galois representations to some eigenclasses in these cohomology, we obtain a bridge between some new classes of automorphic representations and Galois representations. In the first talk we’ll explain the motivation and consequences of the result. Our work is vastly inspired by an analogous characterization of the cohomology of automorphic local systems on locally symmetric spaces proved by Franke, and we’ll also talk about the relation and difference between the 2 situations. In the second talk we’ll sketch the proof and explain some main techniques.

Title:

Abstract: In these lectures we will give an overview on Landau-Ginzburg models, in particular Landau-Ginzburg orbifolds, and matrix factorizations. The starting point is an isolated singularity given by a polynomial $W$ in $n$ variables. We will explain how to associate a (closed) cohomological field theory known as Fan-Jarvis-Ruan-Witten theory or quantum singularity theory to quasihomogeneous singularities. This is the analog of Gromov-Witten theory for projective varieties. We will focus on the Calabi--Yau case.

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.

Title:

Abstract: There has been much recent development on higher symmetries in topological orders as the study of topological phase of maters has become a very active field in condensed matter physics. In this talk, we will carry out the mathematical foundation of a recent joint project with Tian Lan and Xiao-Gang Wen in the above direction. Higher groups are group objects in a higher category. In a very concise way, they can be realised as a simplicial object satisfying suitable Kan conditions. We will give a more explicit algebraic model to realise some of them. This model is between the target of Dold-Kan functor and the skeleton case. Then we discuss higher gauge theory for topological orders, classification of them in lower dimensions, and explicit realisations in special cases.

Bohan Fang (Peking University)

Xiaobo Liu (Peking University)

Speaker:

Motohico Mulase (UC Davis)

Emanuel Scheidegger (Peking University)

Chenchang Zhu (Georg-August-Universität Göttingen)

Schedule:

**10.8 (Monday)**09:30-11:30 Emanuel Scheidegger

11:30-14:00 Lunch Break

14:00-16:00 Motohico Mulase

**10.9 (Tuesday)**09:30-11:30 Motohico Mulase

11:30-14:00 Lunch Break

14:00-16:00 Emanuel Scheidegger

**10.10 (Wednesday)**09:30-11:30 Motohico Mulase

11:30-14:00 Lunch Break

14:00-16:00 Free

**10.11 (Thursday)**09:30-11:30 Emanuel Scheidegger

11:30-14:00 Lunch Break

14:00-16:00 Chenchang Zhu

**10.12 (Friday)**09:30-11:30 Motohico Mulase

11:30-13:00 Lunch Break

13:00-15:00 Emanuel Scheidegger

Title & Abstract:

Title:

**Topological recursion in geometry**(Motohico Mulase)Abstract: 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.The canonical extensions of automorphic vector bundles on Shimura varieties are vector bundles whose global sections characterize holomorphic automorphic forms, and in this series of talks we show that the Hecke module structures on their higher cohomology also come from automotphic forms. As people have associate Galois representations to some eigenclasses in these cohomology, we obtain a bridge between some new classes of automorphic representations and Galois representations. In the first talk we’ll explain the motivation and consequences of the result. Our work is vastly inspired by an analogous characterization of the cohomology of automorphic local systems on locally symmetric spaces proved by Franke, and we’ll also talk about the relation and difference between the 2 situations. In the second talk we’ll sketch the proof and explain some main techniques.

Title:

**Landau-Ginzburg Models and Matrix Factorizations**(Emanuel Scheidegger)Abstract: In these lectures we will give an overview on Landau-Ginzburg models, in particular Landau-Ginzburg orbifolds, and matrix factorizations. The starting point is an isolated singularity given by a polynomial $W$ in $n$ variables. We will explain how to associate a (closed) cohomological field theory known as Fan-Jarvis-Ruan-Witten theory or quantum singularity theory to quasihomogeneous singularities. This is the analog of Gromov-Witten theory for projective varieties. We will focus on the Calabi--Yau case.

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.

Title:

**Higher Groups in Higher Gauge Theory**(Chenchang Zhu)Abstract: There has been much recent development on higher symmetries in topological orders as the study of topological phase of maters has become a very active field in condensed matter physics. In this talk, we will carry out the mathematical foundation of a recent joint project with Tian Lan and Xiao-Gang Wen in the above direction. Higher groups are group objects in a higher category. In a very concise way, they can be realised as a simplicial object satisfying suitable Kan conditions. We will give a more explicit algebraic model to realise some of them. This model is between the target of Dold-Kan functor and the skeleton case. Then we discuss higher gauge theory for topological orders, classification of them in lower dimensions, and explicit realisations in special cases.

This talk also serves as the weekly Symplectic Geometry and Mathematical Physics Seminar.

Contact:

Ms.Ying Hao

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

Tel: 010-62744130

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