1st Year Geometry seminar

We will normally meet on Fridays 5-6:30pm in room 507. We will be mostly following for the first few talks the text "Floer homology groups in Yang-Mills theory" by S.K. Donaldson.

More references:

- Donaldson-Kronheimer, "The geometry of four-manifolds"

- A. Floer, "An instanton invariant for 3-manifolds"

- J.W. Morgan, "The Seiberg-Witten equations and application to the topology of smooth 4-manifolds".


Friday February 8, "Introduction: Interactions between Yang-Mills theory and 3-manifolds, Part 1", Juan Muñoz Echániz

I will first review the basics of Yang-Mills theory for compact 4-manifolds: moduli spaces of instantons, the Yang-mills functional and Uhlenbeck's theorems. However, the main theme of the talk will be the interaction between the instanton theory in 3 dimensions and 4 dimensions, specifically on the tube XxR, motivated by our goal to study ASD connections on 4-manifolds with tubular ends. I will show how this leads to a familiar picture from Morse theory, and introduce the main objects that we will be studying in the next talks.

Thursday February 14 (4:30-6pm), "Yang-Mills theory and 3-manifolds, Part 2", Juan Muñoz Echániz

Continuation of talk from previous week. 

Friday February 15, "Linear Analysis Part I", Yash Deshmukh

We set up the index problem for manifolds with tubular ends and prove the index addition formula. In part I, we shall restrict ourselves to connections which are acyclic over the ends.

Friday March 1, Sam DeHority