报告题目：4-dimensional geometry and topology

报 告 人：Vassily Olegovich Manturov 教授 Moscow Institute of Physics and Technology

报告时间：2023年1月8日-2月5日 19：00-20：30

报告地点：ZOOM ID: 810 4009 0103，PW: 989134

会议链接：https://us02web.zoom.us/j/81040090103?pwd=OTVwM1BBN29Wb1gyL1JrVW40SFg0QT09

校内联系人：Seongjeong Kim kimseongjeong@jlu.edu.cn

Lecture 1. 2023.01.08

The Poincare conjecture h-cobordism theorem

We formulate and proof the famous h-cobordism theorem by Smale

and the Poincare conjecture in dimensions greater than or equal to 5

Lecture 2. 2023.01.13

Freedman's proof of the Poincare conjecture in dimension 4.

We introduce Casson's handles and prove the topological Poincare

conjecture in dimension 4.

Lecture 3. 2023.01.15

Intersection forms.

We discuss algebraic classification of Z-valued quadratic forms.

We shall consider simply connected 4-manifolds and formulate

some classification results based on the intersection pairings on

second homology groups.

Lecture 4. 2023.01.20

Theorems of Wall and Rokhlin

We prove classical theorems of Wall (about stable equivalence of manifolds)

and Rokhlin (about signatures and cobordidsms)

Lecture 5. 2023.01.27

Preliminary materials: spin structures, Clifford algebras

We introduce necessary algebraic background needed for formulation

of Donaldson and Seiberg-Witten invariants

Lecture 6. 2023.01.29

Donaldson's theorem

We sketch the proof of Donaldson's theorem that if an intersection

form of a smooth manifold is positive-definite then it is diagonalisable.

Lecture 7. 2023.02.03

Seiberg-Witten invariants

We give a quick introduction into Seiberg-Witten invariants

and list results: short proof of Donaldson's theorem,

triviality of SW results for positive curvature,

exotic smooth structures

Lecture 8. 2023.02.05

Exotic R^{4}.

We discuss the approaches to constructing smooth structures on R^{4}

due to Freedman, Gompf, Taubes, including the celebrated Taubes'

theorem on continuously many different smooth structures

报告人简介：Vassily Olegovich Manturov，现就职于Moscow Institute of Physics and Technology。研究方向为Low dimensional topology and Knot theory。目前已发表文章150余篇，引用1500余次。担任杂志Journal of Knot Theory and Its Ramifications主编。 出版多本纽结理论相关的书。组织纽结理论相关的国际会议，Three international conferences in the Mathematical Institute (Oberwolfach) on knot theory and low-dimensional topology，International conference ``4-th Russian China Russia-China on Knot theory and Related topics’’ in Bauman State Technical University，A topological seminar "Moscow-Beijing" in Tsinghua University等。