Thomas Walpuski

Seminar: Gauge Theory/Eichtheorie (Wintersemester 2020-21)

In the winter semester 2020-21, I will run a seminar on gauge theory at Humboldt-Universität 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.
  1. geometric foundations:
    1. Vector bundles, connections, curvature
    2. Characteristic classes, Chern–Weil theory
    3. Yang–Mills functional, ASD instantons, the BPST instanton
  2. analytic foundations:
    1. Sobolev spaces and elliptic theory
    2. Uhlenbeck's gauge fixing theorem
    3. Uhlenbeck's removable singularities
    4. Fredholm theory and Banach manifolds
    5. Taubes' gluing construction
    6. Compactness, bubbling of instantons
  3. Donaldson's diagonalization theorem:
    1. Overview of the topology of 4-manifolds, statement of the diagonalization theorem
    2. Proof of Donaldson's diagonalization theorem (multiple talks)
  4. Possible further topics:
    1. Taubes' construction of the Casson invariant
    2. Price's monotonicity formula and compactness in Yang–Mills theory in higher dimensions
    3. Tao–Tian's removable singularities of Yang–Mills connections in higher dimensions
    4. Gauge theory and complex geometry, stable bundles and the Donaldson–Uhlenbeck–Yau Theorem