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.
- 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".
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.
Continuation of talk from previous week.
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.
We extend the theory to include adapted bundles which may not be acyclic over the ends by introducing weighted Sobolev spaces. We extend the index addition formula to this context and study how the index varies with the weights.
We conclude the index theory by proving an index formula for a 4-manifold with tubular ends with cross-sections given by homology 3-spheres. This will relate (mod 8) the index of an adapted bundle to topological invariants of the base space. Finally, we discuss how the L^2 theory developed so far extends to L^p.