Program
Student Seminar On Higgs Bundles and Hitchin Systems
Student No.:20
Time:Wed 19:00-21:00, Oct.10-Jan.16
Instructor:Eduard Looijenga, Bin Wang, Zhiwei Zheng  
Place:Conference Room 1, Jin Chun Yuan West Bldg.
Starting Date:2018-10-10
Ending Date:2019-1-16

2018-12-5

Speaker: Xueqing Wen

Title: Spectral Curves and Generalized Theta Divisors

Abstract: I will talk about spectral curves first, then its application to the moduli spaces of semi-stable vector bundles over curves, including Hitchin maps and generalized theta divisors.


2018-11-28

Speaker: Xun Lin

Title: Moduli Space of Stable Sheaves on K3 Surfaces

Abstract: In this talk, I will talk about the moduli space of stable sheaves on K3 surfaces. Firstly, I will recall some facts of coherent sheaves and define Gieseker (semi-) stability which is relevent to GIT theory. Next I will focus on K3 case. Due to Mukai, there are nice structures on moduli space of stable sheaves. Finally, I will give some examples of low dimension, and the relationship with Hilbert scheme of finite points.


2018-11-21

Speaker: Zhiwei Zheng

Title: Global topology of moduli of Higgs bundles

Abstract: This time I will concentrate on the topology of moduli of Higgs bundles. I will first survey the history and discuss the motivations of understanding Higgs moduli space. I will show the cohomology of Higgs moduli space is pure and concentrating in low degrees. A core conjecture (made in 2008) is by Hausel and Rodriguez-Villegas on vanishing of certain intersections, which is proved by Heinloth in 2016. We will have more talks on this.


2018-11-14

Speaker: Bin Wang

Title: Introduction to Hitchin fibration II

Abstract: In This talk, we will talk about the construction of spectral curves and give explicit BNR correspondence between higgs bundles on base curves and line bundles on spectral curves in the cases GL_{n}, Sp(2n), SO(2n), and SO(2n+1). These show that Hitchin map gives rise to algebraically integrable systems.


2018-11-7

Speaker: Bin Wang

Title: Introduction to Hitchin fibration I

Abstract: In This talk, we will talk about the construction of Hitchin fibration, and give a sketch proof of Poisson commutativity and then show that it is an algebraically complete integrable system for vector bundles. The main reference is Hitchin's paper.


2018-10-31

Speaker: Yunpeng Zi

Title: Moduli of Vector Bundles and GIT in a Nutshell II

Abstract: In This talk, I will give a brief summary of moduli spaces of semi-stable vector bundles of the fixed rank and degree with necessary proof.


2018-10-24

Speaker: Yunpeng Zi

Title: Moduli of Vector Bundles and GIT in a Nutshell

Abstract: In this talk, I will give a brief summary of GIT theory. As an application, I will talk about the moduli of semi-stable vector bundles and semi-stable Higgs bundles on a smooth projective curve.


2018-10-17

Speaker: Xiaoyu Su

Title: Prelimminary to Simpson correspondence over complex numbers II

Abstract: In this talk, I will give a sketch of the proof for the main theorems in non-abelain hodge theory.


2018-10-10

Speaker: Xiaoyu Su

Title: Preliminary to Simpson correspondence over complex numbers

Abstract: In this talk, I intend to give an introduction to the classical Non abelian Hodge theory. State the main theorem and introduce the history of the theory. If we have enough time, I will also talk about some applications of the theory. e.g. the relation between variation of Hodge structure and fixed points in the moduli space under the $\G_m$-action,rigid connections and so on.