**Symposium| Institute of Mathematical Sciences**

**Organizer：**Zheng Zhang

**Date：**Saturday, September 17th, 2022

**Location：**R408, IMS

**Schedule**

**9:30-10:20**

Titile: On the stratifications of moduli space of vector bundles

Speaker: Lingguang Li

Abstract: I will talk about the Shatz and Bialynicki-Birula stratifications of the moduli space of Higgs bundles, and the Frobenius stratification of moduli space of vector bundles in positive characteristic.

**10:30-11:20**

Titile: A Gersten complex on real schemes

Speaker: Fangzhou Jin

Abstract: We discuss a connection between coherent duality and the Verdier-type duality on real schemes via a Gersten-type complex. This is a joint work with H. Xie.

**13:30-14:20**

Titile: Gaeta resolutions and strange duality over rational surfaces

Speaker: Yinbang Lin

Abstract: We will discuss about resolutions of coherent sheaves by line bundles from strong full exceptional sequences over rational surfaces. We call them Gaeta resolutions. We then apply the results towards the study of the moduli space of sheaves, in particular Le Potier's strange duality conjecture. We will show that the strange morphism is injective in some new cases. One of the key steps is to show that certain Quot schemes are finite and reduced. The next key step is to enumerate the length of the finite Quot scheme, by identifying the Quot scheme as the moduli space of limit stable pairs, where we are able to calculate the (virtual) fundamental class. This is based on joint work with Thomas Goller.

**15:00-15:50**

Titile: A Remark on local Polar Multiplicities

Speaker: Xiping Zhang

Abstract: The local polar varieties and their multiplicities were introduced by Teissier to study the algebraic nature of Whitney stratification. It was later shown that they are closely related to the other topological invariants of stratified spaces, such as the local Euler obstructions. In this talk we will discuss some applications via the theory of polar varieties, mostly in the case of hypersurfaces. This is a joint work with Terence Gaffney.

**16:00-16:50**

Titile: Non-abelian Hodge correspondence and the P=W conjecture

Speaker: Zili Zhang

Abstract: Fix a complex projective curve C and a reductive group G. There are three moduli spaces with the pair (C,G): the character variety M_B, the moduli of flat connections M_dR, and the moduli of Higgs bundles M_Dol. The non-abelian Hodge correspondence says there are natural homeomorphisms among the three moduli spaces, and hence identify the cohomology groups of them. The geometric structures of the moduli spaces induce various filtrations in the cohomology groups. De Cataldo-Hausel-Migliorini conjectured in 2012 that the Perverse filtration (P) of M_Dol is identical to the Hodge-theoretic weight filtration (W) of M_B; the P=W conjecture. We will introduce the background and recent progress of the nonabelian Hodge correspondence and the P=W conjecture. The talk is not aimed at specialists.