Topics in Group Theory
This is an original introductory course on Group Theory based on examples. Group Theory is a rigorous way to use the concept of symmetry in Mathematics with wide range of applications. The course addresses a diverse audience (of undergraduate students) with diverse background. We introduce basic concepts and methods of group actions, using motivation from Geometry and Number theory. We apply those results to studying structure of finite groups, and illustrate them with examples. This course is aiming at helping the students to see how concept of symmetry connects different areas of Mathematics, and to develop their individual research interests.
讲师
日期
2024年03月07日 至 05月31日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周四 | 13:30 - 15:05 | A3-1-103 | ZOOM 11 | 435 529 7909 | BIMSA |
| 周五 | 09:50 - 11:25 | A3-1-103 | ZOOM 11 | 435 529 7909 | BIMSA |
修课要求
basic Algebra
课程大纲
1. Definition of a group and main examples. Maps X --> X and permutations. Linear transformations and GL(n,R). Additive and multiplicative groups modulo natural n.
2. Co-sets and normal subsgroups. Quotient groups. Examples: (R,+)/(Z,+); S_n/A_n; O(n,R)/SO(n,R); (Z,+)/(nZ,+)
3. Orders of group elements. Cyclic (sub)groups: finite and infinite. Fundamental theorem on cyclic groups.
4. Group homomorphisms. Isomorphism theorems. Direct products. Chinese Remainder Theorem.
5. Definition of group action. Examples. Action on co-sets. Cayley's Theorem.
6. Adjoint action. Conjugation action on subgroups.
7. The symmetric group. Cycle decomposition and Young Tableaux. Partition number and Ramanujan's congruences. Abelian p-groups..
8. Automorphism groups. Isometry groups.
9. Semidirect products. Sylow Theorems.
10. Finitely generated Abelian groups. Smith normal form. Fundamental Theorem.
11. Group extensions. Central extensions. The first and the second cohomology groups.
2. Co-sets and normal subsgroups. Quotient groups. Examples: (R,+)/(Z,+); S_n/A_n; O(n,R)/SO(n,R); (Z,+)/(nZ,+)
3. Orders of group elements. Cyclic (sub)groups: finite and infinite. Fundamental theorem on cyclic groups.
4. Group homomorphisms. Isomorphism theorems. Direct products. Chinese Remainder Theorem.
5. Definition of group action. Examples. Action on co-sets. Cayley's Theorem.
6. Adjoint action. Conjugation action on subgroups.
7. The symmetric group. Cycle decomposition and Young Tableaux. Partition number and Ramanujan's congruences. Abelian p-groups..
8. Automorphism groups. Isometry groups.
9. Semidirect products. Sylow Theorems.
10. Finitely generated Abelian groups. Smith normal form. Fundamental Theorem.
11. Group extensions. Central extensions. The first and the second cohomology groups.
参考资料
1) Standard textbooks covering finite groups, for example:
W.Ledermann, Introduction to Group Theory, Longman, 1973
J.F.Humphreys, A course in group theory, Oxford, 1996
M.A.Armstrong, Groups and Symmetry, Springer, 1988
2) Further reading:
J.Carrell, Groups, Matrices and Vector Spaces: A Group Theoretic Approach to Linear Algebra, Springer, 2017
E.Artin, Geometric Algebra, Princeton University Press, 1957
W.Ledermann, Introduction to Group Theory, Longman, 1973
J.F.Humphreys, A course in group theory, Oxford, 1996
M.A.Armstrong, Groups and Symmetry, Springer, 1988
2) Further reading:
J.Carrell, Groups, Matrices and Vector Spaces: A Group Theoretic Approach to Linear Algebra, Springer, 2017
E.Artin, Geometric Algebra, Princeton University Press, 1957
听众
Undergraduate
, Advanced Undergraduate
, Graduate
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
谢尔盖·奥布莱津（Sergey Oblezin）于2004年在莫斯科物理技术学院（MIPT）获得博士学位。他在莫斯科接受的教育以及在阿利哈诺夫理论与实验物理研究所（ITEP）的工作经历，塑造了他独特的跨学科视野——以量子物理与数学之间相互启发、彼此转化的深刻融合为基础。
他的早期研究成果获得了多项荣誉，包括两次俄罗斯联邦总统青年数学家奖学金（2007–2008年和2008–2009年）。2009至2012年，他的研究荣获皮埃尔·德利涅奖（Pierre Deligne Prize，由德利涅2004年巴尔赞奖资助设立）。2013至2017年，他主持的项目“拓扑场论、Baxter算子与朗兰兹纲领”获得英国工程与自然科学研究理事会（EPSRC）“成熟职业阶段”（Established Career）研究基金支持。
2015至2023年，谢尔盖任英国诺丁汉大学几何学副教授，2024年全职加入北京雁栖湖应用数学研究院（BIMSA），担任教授。他长期致力于将量子物理中的方法与构造引入并发展于朗兰兹纲领的研究。其研究兴趣包括表示论、调和分析，以及它们与数论和数学物理的深刻联系。