Some thoughts on mathematics

There is an upcoming conference celebrating Cameron Gordon's 80th birthday.  Cameron has had a big influence on my career (both directly and through his mathematics), so I wanted to write some posts giving quick summaries of his influential work.  These will be short samples instead of giving full details on anything.  This post was inspired by a comment of Josh Wang. I thought it might be kind of cliche to start with the most famous results, like the Knot Complement Theorem, so I'll instead start with one of my personal favorites: "Only integral surgeries can yield reducible manifolds" by Gordon-Luecke.  While the title is the result, we'll see some more background and at the end, I'll give a really nice application to the unknotting number of knots.   First, we need to have a little chat about Dehn surgeries.  (As an aside, Cameron is also a historian of topology.  Since Dehn came up, I'll point out Cameron wrote a historical arti...

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...