2020-2021年度西部几何拓扑工作营
短期课程
Title：示性类的Chern-Weil理论
Speaker: 冯惠涛（南开大学）
Abstract: TBA
CW理论ch1-3.pdf
Title：Toroidal geometry
Speaker: 胡正宇（重庆理工大学）
Abstract: TBA
Prerequisites：Hartshorne《代数几何》前3章
weak toroidalisation.pdf
weak ss-reduction in char 0.pdf
Title：Introduction to elliptic genus
Speaker: 韩飞（新加坡国立大学）
Abstract: TBA
Intro to ell-2021.01.16.pdf
Intro to ell-2021.01.17.pdf
Intro to ell-2021.02.01.pdf
Title：Introduction to rational homotopy theory
Speaker: 黄瑞芝（中国科学院）
Abstract: TBA
Title：Some topics on characteristic classes and Schubert calculus
Speaker: 李平（同济大学）
Abstract: 这是一门研究生阶段的短课，我们计划围绕Pontrjagin class来介绍一些cobordism theory和genus, 围绕Chern classes来介绍complex Grassmannians上的Schubert calculus。如果时间允许，我们还会涉及一点其他相关话题。
Prerequisites：听众需要具有同调论的基本知识和一点示性类的概念（比如公理化定义的存在性）
Title： Introduction to Quantization
Speaker: 李思（清华大学）
Abstract: We give an introduction to basic ideas and various recent mathematical developments about quantization that arises from quantum field theory and string theory. Several applications to geometry and topology will be discussed along the journey.
Lecture 1-2021.01.18.pdf
Lecture 2-2021.01.19.pdf
Title：Twisted Vafa-Witten invariants and S-duality
Speaker: 蒋云峰（KU）
Abstract: For a real four manifold M, the S-duality conjecture of
Vafa-Witten (1994) predicts that the S-transformation sends the gauge group SU(r)-invariants counting instantons on M to the Langlands dual gauge group SU(r)/Z_r-invariants counting SU(r)/Z_r-instantons on M; and both of the invariants satisfy modularity properties. This is a generalization of electro-magnetic duality in physics. On mathematics side the SU(r)-Vafa-Witten invariants have been constructed by Tanaka-Thomas using the moduli space of semistable Higgs bundle or sheaves on a smooth complex projective surface underlying M. In this talk I will present the idea of using moduli space of twisted sheaves and twisted Higgs sheaves on a projective surface to define the Langlands dual gauge group SU(r)/Z_r-Vafa-Witten invariants, and provide the proposal to prove the S-duality conjecture of Vafa-Witten for algebraic surfaces. A particular case of K3 surface is proved by explicit calculation.
S-duality_1_virtualEuler-2021.01.26.pdf
S-duality_2_3-2021.01.27.pdf
Title: Canonical sections of Hodge bundles
Speaker: 沈洋（南京大学）
Abstract: In this talk, we introduce our recent work on the canonical sections of Hodge bundles. First, we review the work of the sections of Hodge bundles for Calabi-Yau manifolds, which uses the method of deformation theory. Then we generalize it to the Calabi-Yau type case, using the method of Hodge theory. Finally, we introduce the applications to characterizing the moduli spaces of certain polarized manifolds as ball quotients.
Title： Lectures on complex Finsler geometry.(8h)
Speaker: 万学远（韩国高等研究院）
Abstract: In these lectures, we will recall some basic definitions and facts in complex Finsler geomtry firstly, and then we will show the representation of Chern classes by using complex Finsler metric, the construction of Donaldson type functional in Finsler setting, the existence of Finsler-Einstein and Hermitian-Einstein metrics. And the poisson-Kahler forms on projective bundles, holomorphic sectional curvature of complex Finsler metrics, and dual complex Finsler metrics are discussed.
Lectures on complex Finsler geometry v1.pdf
Title：Lectures on hyperbolic surfaces
Speaker: 吴云辉（清华大学）
Abstract: TBA
Title：Introduction to deformation of complex structures
Speaker: 夏炜（中山大学）
Abstract: I will first introduce some basics about Kodaira-Spencer's deformation theory of complex manifolds, then I will describe a new proof of a theorem of Clemens which says that ambient cohomology of a Kahler manifold annihilates obstructions.
Introduction to deformation of complex structures.pdf
Title：从麦克斯韦方程到广义相对论
Speaker: 徐浩（浙江大学）
Abstract: TBA
相对论讲稿.pdf
Title：Kähler流形的几何与拓扑
Speaker: 杨晓奎（清华大学）
Abstract: TBA
Title：Stability, moduli space and wall crossing
Speaker:张诗卓(The University of Edinburgh)
Abstract: I will introduce a series recent work on Bridgeland moduli spaces of stable objects in the Kuznetsov components of prime Fano threefolds including our recent work on disproving the Kuznetsov's Fano threefold conjecture. In my talk, I focus on various techniques to prove stability of some objects in different settings. On the first day, I will introduce classical stability. Then I give a brief survey on Bridgeland stability conditions on surfaces and higher dimension varieties. I will compute walls for several characters and discuss wall crossing in these cases.
On the second day, I introduce the basic materials on derived category of coherent sheaves on smooth projective varieties, semi-orthogonal decompositions and various operations including left and right mutations. Then I will introduce the concept of the Kuznetsov components of prime Fano threefolds and construct stability conditions on it. On the third day, I will construct several bridgeland moduli space in the Kuznetsov component of prime Fano threefold of index two, following the work of Altavilla-Petkovič-Rota, Pertusi-Yang, Bayer-Beentjes-Feyzbakhsh-Hein-Martinelli-Rezaee-Schmidt. On the last day, I will construct several bridgeland moduli space in prime Fano threefold on index one, following my recent work and our work with Jacovskis and Lin. I will talk about my recent result on Kuznetsov's Fano threefold conjecture and refined (birational)categorical Torelli for Gushel-Mukai threefold.
Stability, moduli space and wall crossing-1.pdf
Stability, moduli space and wall crossing-2.pdf
