【开课说明】
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
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【报名流程及联系人】
1. 通过Email提交报名申请，附件报名表电子版见本页面底部
（文件命名格式为“20200706蒋云峰短期课程报名表+学员姓名+单位名”）；
2. 报名邮件发至：dxmeisx@126.com，邮件主题为“20200706蒋云峰短期课程报名申请”；
3. 报名截止日期为 2020年07月03日，以电子邮件收信时间为准。
联系人：邓老师 电话：13469861099
Email: dxmeisx@126.com
【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
