Institute of Mathematical Sciences
Time：4:00pm-5:00pm, July 16, Monday
Location：Room 302, Library
Speaker: Jiang Yunfeng
Jiang Yunfeng is currently an Associate Professor at University of Kansas.He got my Ph.D from University of British Columbia in 2007, and after that was a Postdoc at University of Utah and Imperial College London.Prof. Jiang Yunfeng is working in the area of Algebraic Geometry, more precisely Gromov-Witten and Donaldson-Thomas theory motivated by string theory and Gauge theory.
蒋云峰，美国堪萨斯大学数学系副教授。蒋云峰教授的主要研究方向：代数几何，以弦论和规范理论为基础的Gromov-Witten和Donaldson-Thomas理论。
Title:MacPherson's index theorem and Donaldson-Thomas invariants
Abstract:The Gauss-Bonnet-Chern theorem states that for a smooth compact complex manifold, the integration of the top Chern class over the manifold is its topological Euler characteristic. In order to study Chern class for singular varieties or schemes, R. MacPherson introduced the notion of local Euler obstruction of singular varieties. The local Euler obstruction is an integer value constructible function.A characteristic class for the local Euler obstruction was defined by using Nash blow-ups, and is called the Chern-Mather class or Chern-Schwartz-MacPherson class. The Chern-Schwartz-MacPherson class can be used to define Chern class for singular varieties.
Inspired by gauge theory in higher dimension and string theory, the curve counting theory via stable coherent sheaves was constructed by Donaldson-Thomas on projective 3-folds, which is now called the Donaldson-Thomas theory. In the case of the Calabi-Yau threefolds, the Donaldson-Thomas invariants are proved by Behrend to be weighted Euler characteristic of the moduli space, where the weights come from the local Euler obstruction of the moduli space. In this talk I will discuss some results of the Donaldson-Thomas invariants along this line, and talk about one case that how the Behrend weighted Euler characteristic is related the Y. Kiem and J. Li's cosection localization invariants. If time permits, I will also talk about the formal and non-archimedean version of the moduli space (stack) of stable coherent sheaves and its relation to motivic Donaldson-Thomas invariants.
题目：MacPherson指标定理和Donaldson-Thomas不变量
摘要：Gauss-Bonnet-陈省身定理指出，对于光滑紧致的复流形，陈类的积分是其拓扑欧拉示性数。为了研究奇异流形或解析簇的陈示性类，R.MacPherson引入了局部欧拉障碍的概念。局部欧拉障碍是一个整数值可构造函数。局部欧拉障碍的一个特征类是用Nash blow-up定义的，被称为“Chern-Mather”或“Chern-Schwartz-MacPherson”类。Chern-Schwartz-MacPherson类可以用来为奇异的流形定义陈类。
受高维度的规范理论和弦理论的启发， Donaldson-Thomas在射影3-流形上构造了曲线计数理论，这一理论现在被称为“Donaldson-Thomas理论”。在卡拉比-丘流形的情况下，Behrend证明 Donaldson-Thomas不变量是模态空间的“加权欧拉示性数”，其中的权重来自于模空间的局部欧拉障碍。在这次讲座中，Donaldson-Thomas不变量的一些结果，并讨论一个案例，即Behrend加权欧拉示性数是如何与Y.Kiem和李俊的cosection局部化不变量相关的。如果时间允许，我还将讨论Donaldson-Thomas模空间的非阿基米德域的版本，以及它与motivic Donaldson-Thomas不变的量关系。