TGMRC专题网课：GW-DT-VT理论

       2020年7月6日北京时间10；00-11:15，美国堪萨斯大学蒋云峰教授讲授的“Course on Gromov-Witten, Donaldson-Thomas and Vafa-Witten Theory”课程正式开讲。该专题课程由国家天元数学中部中心和三峡数学研究中心联合组织组织。当天，三峡数学研究中心主任林宗柱教授主持，有60多位来自国内外各大高校的年轻教师和博士研究生、硕士研究生参与了该线上课程学习，课堂交流热烈。

本次课程从7月6日起，每周两次授课，总学时数达30小时。因疫情影响，我们只能采用了网上授课的模式，突破了传统授课模式，但又拓宽了学术交流的方式，使得学术交流更加高效便捷，有效节约学术交流成本，但又使得课程的受众更加广泛。

为了辅助课堂教学，林宗柱教授全程担任本课程助教，每周组织一次习题课。此外，为了方便交流，及时更新课程视频，我们成立了“TGMRC专题网课：GW-DT-VT理论”微信学习群，课后群里讨论热烈，学术氛围浓厚。

以下为本次课程的相关介绍及视频会议相关信息，欢迎相关领域的老师同学参与学习交流。

开课日期：2020年07月06日

举办地点：线上课程（zoom会议）

授课时间：7月6日（含）起 每周一、周四上午 10:00-11:15 课程持续十周

周一课程会议ID：815 4998 2426  周四课程会议ID：899 0981 1480  密码：123456

主讲人：蒋云峰 (University of Kansas)  导师简介见本页面底部附件

主办单位：国家天元数学中部中心、三峡数学研究中心

【ABSTRCT】

This course is an introduction to the theories of counting invariants in modern enumerative geometry, which includes Gromov-Witten, Donaldson-Thomas, and more recent Vafa-Witten invariants. We first review some foundational work of prefect obstruction theory following Jun Li and Gang Tian, Behrend-Fatechi. We will talk about the basic notion of normal cones, intrinsic normal cones and virtual fundamental classes. We then apply the construction to define Gromov-Witten like invariants and Donaldson-Thomas invariants.

A special but very important case is the ``Symmetric obstruction theory”, which was defined by Behrend and used to prove that Donaldson-Thomas invariants are motivic invariants. We will state the basic idea of the proof. Inspired by the notion of p-feild by Huai-Liang Chang and Jun Li in Gromov-Witten theory and cotangent theory in physics of Cosetllo, we will also talk about the work of signed Euler characteristics of Jiang-Thomas, and its applications to define the Vafa-WItten invariants for algebraic surfaces for both the gauge group SU(r) and its Langlands dual gauge group SU(r)/Z_r. Applications of the Vafa-Witten theory will be given to prove the S-duality conjecture of Vafa-Witten inspired by N=4 supersymmetric Yang-Mills theory in physics.

【OUTLINE】

1. Introduction: outline of the course

2. Basic deformation and obstruction theory

3. Normal cone and intrinsic normal cone, Virtual classes

4. Deformation and obstruction of Gromov-Witten theory: known results and open problems

5. Symmetric obstruction theory

6. Behrend function and Donaldson-Thomas invariants

7. Virtual signed Euler characteristic

8. Vafa-Witten invariants

9. S-duality conjecture

10. S-duality conjecture II