四维流形的拓扑

任课教师 Speaker:林剑锋
时间 Time: 每周三、五 9:50-11:25 2021-9-13 ~ 12-3
地点 Venue:西楼第三会议室

课程描述 Description

Smooth 4-manifolds are important objects in low dimensional topology. This lecture series will introduce 4-manifolds from the following perspectives:

 

(1) classical invariants of 4-manifolds.

(2) Freedman's classification of simply-connected 4-manifolds (without proof).

(3) construction of 4-manifolds (Kirby calculus, surgery, rational blow-down).

(4) the Seiberg-Witten invariants and the Bauer-Furuta invariants of 4-manifolds.

(5) symplectic 4-manifolds.

(6) Donaldson's diagonalizability theorem.

(7) geography and botany problem of smooth 4-manifolds.

(8) exotic phenomena in dimension 4.

(9) embedded surfaces in 4-manifolds, the Thom conjecture and the Milnor conjecture.

(10) Khovanov homology and its application to 4-manifolds.

(11) (time permitting) more recent developments (e.g. Gabai's light bulb theorem)

预备知识 Prerequisites

Basic algebraic topology and differential topology.

参考资料 References

(1) Ronald Fintushel and Ronald Stern, “Six Lectures on 4-manifolds”

(2) John Morgan, “The Seiberg-Witten equations and Applications to the Topology of Smooth Four-manifolds”

(3)  Simon Donaldson and Peter Kronheimer, “The Geometry of Four-Manifolds”

(4)  Robert Gompf and Andras Stipsicz, “4-manifolds and Kirby Calculus”

(5)  Lecture notes from Ciprian Manolescu's class “4-dimensional topology” at Stanford. (Notes written by Shintaro Fushida-Hardy)

Tencent Meeting ID:705 7478 7470

Password:118118

Wechat group: QR code

Office Hours: Wednesdays 2:00-3:00PM, 静斋309 or Tencent meeting (same ID)


 




Lecture Notes:

Lecture 1: Why dimension 4 is special?

Lecture 2: Classical invariants of 4-manifolds