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".


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.


In this lecture we return  to our discussion of instantons over manifolds with tubular ends. After proving exponential decay for finite energy instantons, we will construct a suitable moduli space of instantons. If time permits, we start discussing the non-linear version of additivity property proved in previous lectures.
We describe a glueing result for instantons over adapted bundles, which generalises the addivity property from previous talks.
We will briefly discuss the construction of the moduli spaces of instantons of finite energy over manifolds with tubular ends in the remaining case where ends are non-degenerate. We will then proceed to discuss convergence results and compactness properties of these moduli spaces in the case of tubes. 

We finish the discussion of compactness results and glueing (Juan and Sam).


We shall motive Floer's construction of Instaton homology and continuation maps by discussing the these in the Morse homology case. After this we will move to the construction of Instaton homology over Z/2.


We will review how moduli of instantons can be compactified by adding chains of `ideal instatons'. We then construct and use the continuation maps to show invariance of the instaton homology under deformation of the metric. If time permits we start discussing the orientations of the moduli spaces, needed to construct the instaton homology with coefficients in Z