Some thoughts on mathematics

In this post, I'll try to describe a little bit about moduli spaces of ASD connections on closed four-manifolds.   This includes a very rough sketch of Donaldson's diagonalization theorem. Much of this is a summary of content in Saveliev's book "Invariants for Homology 3-Spheres".   Four-manifolds:  On three-manifolds, we were interested in flat connections , which roughly correspond with representations of the fundamental group.  Much of the excitement of four-manifold topology revolves around simply-connected four-manifolds, so this would not always be an effective invariant.    Instead, we work instead with what is called the ASD moduli space.  Let $X$ be a closed, connected, oriented, smooth 4-manifold and $E$ a principal $SU(2)$ bundle (these are determined by $\langle c_2(E), [X] \rangle$, so you can just think of a choice of integer), we can consider the solutions to the so-called  anti-self-dual (ASD) equation :   $$F_A...

The goal of this sequence of posts is to talk about instantons and sketch the proof of a recent paper with Ali Daemi and Mike Miller Eismeier ( https://arxiv.org/abs/2207.03631 ) .   I think most mere mortal mathematicians have math phobias.  Here I don't mean a fear of giving talks or people finding out you don't know as much as you think you are tricking them into thinking you know, although each of those fears warrants its own blog post.  Instead I mean being afraid of mathematical concepts.  Earlier in my career I was really afraid of the knot Floer complex, $CFK^\infty$.  The pictures people drew never made sense to me, and I could not envision how anyone could wield this abstract nonsense in a way that would produce topology.  I didn't really make progress on this until I was a postdoc and was collaborating with Jen Hom and going through some old Ozsváth-Szabó papers with Allison Miller.  Sometimes a phobia is going from definitions to ...

 A natural question in topology is whether given two manifolds $N^n$ and $M^m$ with $m > n$, does $N$ embed in $M$.  There's an interesting history to this problem in a variety of dimensions, but I'll focus on the case of embeddings of homology 3-spheres into $S^4$ due to recent work of Clayton McDonald.   In the topological category, Freedman proved that every homology 3-sphere embeds topologically into $S^4$.  Indeed, every homology sphere bounds a contractible topological 4-manifold.  Doubling this contractible 4-manifold produces a copy of $S^4$, since a simply-connected 4-manifold with the same homology as $S^4$ is homeomorphic to $S^4$ by Freedman's classification of closed, simply-connected, topological 4-manifolds.     The smooth case is more interesting, as some homology 3-spheres do not embed smoothly in $S^4$, while some do.  From now on, we work only in the smooth category.  Of course $S^3$ embeds smoothly in $S^4...