Seminar: Gauge Theory/Eichtheorie (Wintersemester 202021)
In the winter semester 202021, I will run a seminar on gauge theory at HumboldtUniversität zu Berlin. 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.
If you have any questions about this seminar that are not answered on this page, then feel free to email me at thomas@walpu.ski
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 is a tentative plan/list of topics for the seminar. As we approach the beginning of the semester, I will add references to the literature.
geometric foundations:
 Vector bundles, connections, curvature
 Characteristic classes, Chern–Weil theory
 Yang–Mills functional, ASD instantons, the BPST instanton

analytic foundations:
 Sobolev spaces and elliptic theory
 Uhlenbeck's gauge fixing theorem
 Uhlenbeck's removable singularities
 Fredholm theory and Banach manifolds
 Taubes' gluing construction
 Compactness, bubbling of instantons

Donaldson's diagonalization theorem:
 Overview of the topology of 4manifolds, statement of the diagonalization theorem
 Proof of Donaldson's diagonalization theorem (multiple talks)

Possible further topics:
 Taubes' construction of the Casson invariant
 Price's monotonicity formula and compactness in Yang–Mills theory in higher dimensions
 Tao–Tian's removable singularities of Yang–Mills connections in higher dimensions
 Gauge theory and complex geometry, stable bundles and the Donaldson–Uhlenbeck–Yau Theorem