The Geometry of Four-Manifolds (Summer 2021)
This is the reading group webpage on Donalson & Kronheimer's book The Geometry of Four-Manifolds. Meetings are three times a week on MWF 10:00~11:00am EST. Mondays are discussions and Wednesday/Fridays are student presentations. The first meeting will be on May 31st (M) and the last one will be on August 11st (W). Please email me at qiuyu_ren@berkeley.edu or message me if you want to present in a future talk or if you have any additional questions.
Professor Tomasz Mrowka will attend our weekly discussion sections to shed some light on our readings. Many thanks to Tom!
I am maintaining an errata list for the book.
Zoom ID: 96751739151
Schedule:
May 31: Introduction. Overview by Professor Mrowka. Chapter 1. Notes
Jun 2: Dashen: A concise introduction to connections on vector bundles and principal bundles; ASD connections. Chapter 2.1.
Jun 4: Dashen: Introduction to Coulomb gauge; skectch proof to Uhlenbeck's theorem and its applications. Chapter 2.3 page 53~58.
Jun 7: Discussion. (Note: Professor Mrowka will be away)
Jun 9: Dashen: Proof of Uhlenbeck's theorem. Chapter 2.3 page 60~69. Notes
Jun 11: Dashen: Finish of proof of Uhlenbeck. Jiakai: Introduction to Chapter 3.
Jun 14: Discussion.
Jun 16: Cancelled.
Jun 18: Jiakai: Spin structures over a point; Dirac operators. Chapter 3.1.
Jun 21: Discussion.
Jun 23: Jiakai: Fourier transform of ASD connections over the four-torus. Chapter 3.2.
Jun 25: Jiakai: Fourier inversion. Chapter 3.2.
Jun 28: Discussion.
Jun 30: Jiakai: ADHM construction. Chapter 3.3.
Jul 2: Jiakai: Finishing up ADHM construction; some examples. Chapter 3.3, 3.4, 4.1.
Jul 5: Discussion.
Jul 7: Qiuyu: Straification structure of the total orbit space; local structure of Yang-Mills moduli spaces. Chapter 4.2 page 129~138.
Jul 9: Qiuyu: Transversality. Chapter 4.3.2, 4.3.3, 4.3.5 (prereq: 4.3.1; taken for granted: 4.3.4). Notes
Jul 12: Discussion.
Jul 14: Qiuyu: Compactification. Chapter 4.4.1, 4.4.3 (taken for granted: 4.4.4). There is a gap in the patching argument in Section 4.4.2 (there may be topological obstruction to extending the gauge transformations). I will sketch another proof (which only works for ASD connections) following Secton 3 of Uhlenbeck's paper. Notes
Jul 16: Qiuyu: Rational cohomology of the orbit space of connections. Chapter 5.1.1, 5.1.2, probably 5.1.4.
Jul 19: Discussion.
Jul 21: Jiakai: Some K-theory; determinant line bundles. Chapter 5.1.3, 5.2.1, probably beginning of 5.4.
Jul 23: Qiuyu: Links of the reducible connections; codimension-two submanifolds. Chapter 5.1.4, 5.2.2.
Jul 26: Discussion. (Note: Professor Mrowka will be away)
Jul 28: Jiakai: Orientation of moduli spaces. Chapter 5.4. Some additional topics on orientation may be included, see Lecture 22,23 of these transcribed notes by Donghao Wang.
Jul 30: Cancelled.
Aug 2: Discussion. Excision and gluing results by Professor Mrowka. Chapter 7.
Aug 4: Mary: Definite forms: proof of Theorem 1.3.1. Chapter 8.1.
Aug 6: Mary: Structure of the compactified moduli spaces. Chapter 8.2.
Aug 9: Discussion.
Aug 11: Mary: Even forms with $b^+=0,1,2$: proof of Theorem 1.3.2. Chapter 8.3.