**Reporter: **Aleksander Doan (University of Cambridge & University College London)

**Time: **15:00-17:00, Aug 7-16 (every Monday, Wednesday, Friday), 2023

**Place: **Room 2121, the Second Teaching Building

**Abstract: **This short series of lectures will explore the analysis, geometry, and topology of the Yang-Mills equations: a nonlinear generalization of Maxwell’s equations originating in particle physics. After a brief introduction to gauge theory, we will discuss connections between Yang-Mills theory and complex geometry, in particular the seminal work of Atiyah and Bott on the Yang-Mills equations over surfaces, as well as the construction of instantons on the four-sphere using twistor theory. Finally, we will outline some applications of gauge theory to four-dimensional topology. These lectures will be largely self-contained and not assume any background other than basic knowledge of topology and smooth manifolds.

Plan of the lectures (10h total)

1. From electromagnetism to topology

2. Connections and curvature

3. Flat connections

4. Holomorphic vector bundles

5. Algebraic and symplectic quotients

6. Yang-Mills equations over surfaces I

7. Yang-Mills equations over surfaces II

8. Self-duality in dimension four

9. Instantons on the four-sphere

10. Towards topological applications