Some thoughts on mathematics

More gauge theory coming at you.  This post is based on Ruberman's paper "An obstruction to smooth isotopy in dimension 4", and inspired by a conversation with John Baldwin. I want to talk about an idea in gauge theory that is slightly less advertised, not too complicated to describe, and still really cool.  Let's say we have a closed four-manifold $X$ (simply-connected, smooth, oriented, connected, large $b^+_2$, all the good things).  A natural question is how to distinguish self-diffeomorphisms of $X$ up to isotopy?  Freedman and Quinn proved that if two diffeomorphisms of $X$ induce the same map on homology, then they are isotopic through homeomorphisms.  Totally cool, but they may not be isotopic through diffeomorphisms!  It turns out we can use gauge theory to make this distinction.  Get pumped!   First, here's the basic idea.  If we fix a metric $g$ on $X$ and an $SO(3)$-bundle $E$ with non-trivial $w_2$, then recall that we ge...

I am a wheel spinner.  I burn a lot of time on research projects which I have been stuck on for a long time, think about regularly, and still have no new ideas for.  Usually this does not lead to new results nor developing new skills.  Often I'm doing this because I feel it's a time that I "should" be being productive, or because I am dreading refereeing or something else work-related I don't want to do.  But working for the sake of working during a specific time or working enough hours is not always the most productive, and definitely not the most enjoyable.  Of course, everyone has different work habits and schedules they need, and should do what's best for them.  Personally, I tend to think: I could use this 1 hour to a) do an awful-sounding work task, b) do something that would make me feel good (e.g. go for a run, read, etc), or c) spin my wheels.  In the end, I often opt for c) because then "at least I'm working" but at the end I have nothi...