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...