北京雁栖湖应用数学研究院 北京雁栖湖应用数学研究院

  • 关于我们
    • 院长致辞
    • 理事会
    • 协作机构
    • 参观来访
  • 人员
    • 管理层
    • 科研人员
    • 博士后
    • 来访学者
    • 行政团队
  • 学术研究
    • 研究团队
    • 公开课
    • 讨论班
  • 招生招聘
    • 教研人员
    • 博士后
    • 学生
  • 会议
    • 学术会议
    • 工作坊
    • 论坛
  • 学院生活
    • 住宿
    • 交通
    • 配套设施
    • 周边旅游
  • 新闻
    • 新闻动态
    • 通知公告
    • 资料下载
关于我们
院长致辞
理事会
协作机构
参观来访
人员
管理层
科研人员
博士后
来访学者
行政团队
学术研究
研究团队
公开课
讨论班
招生招聘
教研人员
博士后
学生
会议
学术会议
工作坊
论坛
学院生活
住宿
交通
配套设施
周边旅游
新闻
新闻动态
通知公告
资料下载
清华大学 "求真书院"
清华大学丘成桐数学科学中心
清华三亚国际数学论坛
上海数学与交叉学科研究院
BIMSA > Introduction to Gauge theory \(ICBS\)
Introduction to Gauge theory
This course provides an introduction to Gauge theory in mathematics for graduate and advanced undergraduate students. We focus on Donaldson's theory on 4-manifolds, which involves the moduli space of anti-self-dual connections, and related gauge-theoretic invariants in four dimensions. The course begins with a review of connections and curvatures on vector bundles and principal G-bundles. Our aim is to understand the Uhlenbeck compactification of the moduli space, its role in formulating the Donaldson invariants, and the connection to the algebraic counterpart in the moduli problem of algebraic geometry. If time permits, we may discuss more advanced topics from the current research field.
Professor Lars Aake Andersson
讲师
田中祐二
日期
2025年03月14日 至 05月30日
位置
Weekday Time Venue Online ID Password
周五 13:30 - 16:55 A14-203 ZOOM 07 559 700 6085 BIMSA
修课要求
manifolds, vector bundles, principal bundles, and the characteristic classes of them such as the Pontryagin and Chern ones
课程大纲
1. Connections, curvatures, and gauge transformations
2. Yang-Mills connections and anti-self-dual instantons
3. Sobolev spaces, elliptic operators, and linearisation
4. Atiyah-Hitchin-Singer complex, Kuranishi model, and Freed-Uhlenbeck perturbation
5. Reducible connections, orientation issues of the moduli space
6. Uhlenbeck compactness, Taubes' gluing, and Donaldson's theorem
7. Uhlenbeck compactification and Gieseker compactifiation
8. Donaldson invariants, Witten's conjecture, and other related invariants
9. Advanced topics
参考资料
S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford University Press, 1990.
D. S. Freed and K. K. Uhlenbeck, Instantons and four-manifolds, Second edition, Mathematical Sciences Research Institute Publications, 1991.
R. Friedman and J. W. Morgan, Smooth four-manifolds and complex surfaces, Springer-Verlag, 1994.
R. Friedman and J. W. Morgan (eds.), Gauge theory and the topology of four-manifolds, IAS/Park City Mathematics Series, Volume 4, American Mathematical Society, 1998.
C. H. Taubes, Metrics, Connections and gluing theorems. CBMS Regional Conference Series in Mathematics, 89, American Mathematical Society, 1996.
C. H. Taubes, Differential Geometry. Bundles, connections, metrics and curvature, Oxford Graduate Texts in Mathematics, 23, Oxford University Press, 2011.
听众
Advanced Undergraduate , Graduate , 博士后
视频公开
不公开
笔记公开
不公开
语言
英文
讲师介绍
My research interests are primarily centred on Gauge theory within mathematics. Recently, my focus has been on semistable Higgs sheaves on complex projective surfaces and associated gauge-theoretic invariants, employing algebro-geometric methods. However, I also have a strong interest in working within the analytic category.
北京雁栖湖应用数学研究院
CONTACT

No. 544, Hefangkou Village Huaibei Town, Huairou District Beijing 101408

北京市怀柔区 河防口村544号
北京雁栖湖应用数学研究院 101408

Tel. 010-60661855
Email. administration@bimsa.cn

版权所有 © 北京雁栖湖应用数学研究院

京ICP备2022029550号-1

京公网安备11011602001060 京公网安备11011602001060