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Expert in a Dying Field

I don't necessarily think the various areas I discuss below are more or less important than the others, just what I view as their popularity.  Others likely have a different view.  Recently, I was listening to The Beths' "Expert in a Dying Field".  It's a break-up song about all the minutia you learn about a person which no longer feels relevant when the relationship ends.  (Don't worry, I'm not going through a breakup/angst phase.  I just like the song.)  When I listen to it, I often times think about Heegaard Floer homology.   Heegaard Floer homology, developed by Ozsvath and Szabo, exploded in the early 2000s.  It seemed to pack most of the punch of gauge theory while being super computable.  When I started grad school there was this incredible flurry of activity, both in theory building (e.g. bordered Floer homology) and applications (e.g. structure of the knot concordance group, classifying fillable contact structures).  Hot fi...
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SNACKs and slice disks

Here's a short post on a construction I really enjoy.   Theorem: (Kawauchi)  Let $K \subset S^3$ be a strongly negative amphichiral knot (SNACK).  Then $K$ bounds a smoothly embedded disk in a rational homology ball.   Let me first explain the statement and then give a proof.  A knot $K$ is strongly negative amphichiral if there is an orientation-reversing diffeomorphism of $(S^3,K)$ which is also orientation-reversing on $K$ and the fixed point set is two points which sit on $K$.  Sometimes $K$ can be slice, e.g. the square knot is a slice SNACK, but often it is not, e.g. the figure-eight knot.  (The Fox-Milnor condition shows it is not even topologically slice.)   Here is the proof, which is very quick.  First, $X_0(K)$ denote a 0-framed 2-handle attachment to $B^4$ along $K$.  Then $K$ is smoothly slice in this four-manifold - the slice disk is just the core of the 2-handle!  (This applies even if $K$ is not SNACK...
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Absolutely exotic contractible four-manifolds

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Since exotic four-manifolds are always "in", I wanted to talk about a cool construction of exotica I like due to Akbulut-Ruberman.  In their paper, they construct compact contractible four-manifolds which are "absolutely exotic", which means they are homeomorphic, but there is no diffeomorphism between them.  To clarify the terminology, often times, for manifolds with boundary, one talks about "relatively exotic" manifolds, which means that you have a homeomorphism from one four-manifold to another, but the boundary homeomorphism doesn't extend to a diffeomorphism of the four-manifolds.  [For context, it's worth pointing out that long before, we know how to construct lots of non-compact, contractible exotica: namely exotic $\mathbb{R}^4$'s.  In fact, Taubes proved there are uncountably many exotic $\mathbb{R}^4$'s.] I want to describe their work (but with a bit of a Floer homology lean).   For simplicity, let's construct a pair of n...
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