Thomas Walpuski

Seminar: Gauge Theory/Eichtheorie (Wintersemester 2020/2021)

In the winter semester 2020-21, I will run a seminar on gauge theory. Gauge theory is vast topic with a long history and an area of a lot of current research. This seminar covers only a small but important part of this subject. The plan for this seminar is to build up the geometric and analytic foundations of gauge theory in dimension four, and to discuss the proof of Donaldson's diagonalization theorem.

This seminar takes place Tuesdays 9:15–11:00 via Zoom starting 2020/11/03. The Zoom invite can be found in the Moodle page (key: uhlenbeck). Here is this seminar's AGNES listing.

If you have any questions about this seminar that are not answered on this page, then feel free to email me at walpuski@math.hu-berlin.de

Who is this seminar for?

This seminar is aimed at students with an interest in differential geometry, geometric analysis, and/or topology. To participate in the seminar, you should have a good understanding of differential geometry. Ideally, you would already have some knowledge of Sobolev spaces, elliptic theory, and Fredholm operators.

The plan

Here are the notes from the organizational meeting.
10.11.2020
Thomas Walpuski
Bundles, connections, and curvature
lecture notes
17.11.2020
Niklas Martensen
Yang–Mills functional, ASD instantons, and the BPST instanton
Niklas' notes
24.11.2020
Solveig Milena Hepp
Sobolev spaces, Fredholm theory, and elliptic theory
Solveig's notes
01.12.2020
Douglas Schultz
Uhlenbeck's gauge fixing theorem
Douglas' notes
08.12.2020
Shubham Dwivedi
Uhlenbeck's removable singularities theorem
15.12.2020
Naageswaran Manikandan
The compactness problem for ASD instantons
05.01.2021
Milica Đukić
Taubes' gluing theorem
12.01.2021
Adrian Philipp Dawid
Overview of the topology of 4–manifolds
19.01.2021
Joshua Egger
Construction of the moduli space of ASD instantons
26.01.2021
Thomas Walpuski
The collar theorem
02.02.2021
Michael Rothgang
Analysis of reducibles and conclusion of the proof of Donaldson's theorem

References

(Baum, 2014; Freed & Uhlenbeck, 1991; Donaldson, 1983; Donaldson, 1985; Donaldson & Kronheimer, 1990; Donaldson, 1990; Donaldson & Thomas, 1998; Taubes, 1982; Taubes, 1984; Taubes, 1990; Uhlenbeck, 1982; Uhlenbeck, 1982; Uhlenbeck & Yau, 1986)
  1. Baum, H. (2014). Eichfeldtheorie. Eine Einfüshrung in die Differentialgeometrie auf Faserbündeln. In Springer-Lehrbuch Masterclass (2nd revised ed., pp. xiv + 380). Heidelberg: Springer Spektrum.
  2. Donaldson, S. K., & Thomas, R. P. (1998). Gauge theory in higher dimensions. In The geometric universe (Oxford, 1996) (pp. 31–47). Oxford Univ. Press. http://www.ma.ic.ac.uk/ rpwt/skd.pdf
  3. Freed, D. S., & Uhlenbeck, K. K. (1991). Instantons and four-manifolds (Second, Vol. 1). Springer-Verlag. https://link.springer.com/book/10.1007%2F978-1-4613-9703-8
  4. Taubes, C. H. (1990). Casson’s invariant and gauge theory. Journal of Differential Geometry, 31(2), 547–599. https://doi.org/10.4310/jdg/1214444327
  5. Donaldson, S. K., & Kronheimer, P. B. (1990). The geometry of four-manifolds.
  6. Donaldson, S. K. (1990). Polynomial invariants for smooth four-manifolds. Topology, 29(3), 257–315. https://doi.org/10.1016/0040-9383(90)90001-Z
  7. Uhlenbeck, K. K., & Yau, S.-T. (1986). On the existence of Hermitian–Yang–Mills connections in stable vector bundles. Communications on Pure and Applied Mathematics, 39(S, suppl.), S257–S293. https://doi.org/10.1002/cpa.3160390714
  8. Donaldson, S. K. (1985). Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proceedings of the London Mathematical Society, 50(1), 1–26. https://doi.org/10.1112/plms/s3-50.1.1
  9. Taubes, C. H. (1984). Self-dual connections on 4–manifolds with indefinite intersection matrix. Journal of Differential Geometry, 19(2), 517–560. http://projecteuclid.org/getRecord?id=euclid.jdg/1214438690
  10. Donaldson, S. K. (1983). An application of gauge theory to four-dimensional topology. Journal of Differential Geometry, 18(2), 279–315. https://doi.org/doi:10.4310/jdg/1214437665
  11. Taubes, C. H. (1982). Self-dual Yang–Mills connections on non-self-dual 4–manifolds. Journal of Differential Geometry, 17(1), 139–170. http://projecteuclid.org/getRecord?id=euclid.jdg/1214436701
  12. Uhlenbeck, K. K. (1982). Connections with L^p bounds on curvature. Communications in Mathematical Physics, 83(1), 31–42. https://projecteuclid.org/euclid.cmp/1103920743
  13. Uhlenbeck, K. K. (1982). Removable singularities in Yang–Mills fields. Communications in Mathematical Physics, 83(1), 11–29. http://projecteuclid.org/getRecord?id=euclid.cmp/1103920742