報告題目：4-dimensional geometry and topology
報 告 人：Vassily Olegovich Manturov 教授 Moscow Institute of Physics and Technology
報告時間：2023年1月8日-2月5日 19：00-20：30
報告地點：ZOOM ID: 810 4009 0103，PW: 989134
會議鏈接：https://us02web.zoom.us/j/81040090103?pwd=OTVwM1BBN29Wb1gyL1JrVW40SFg0QT09
校內聯系人：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
報告人簡介：Vassily Olegovich Manturov，現就職于Moscow Institute of Physics and Technology。研究方向為Low dimensional topology and Knot theory。目前已發表文章150余篇，引用1500余次。擔任雜志Journal of Knot Theory and Its Ramifications主編。 出版多本紐結理論相關的書。組織紐結理論相關的國際會議，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。
