Some thoughts on mathematics

There's an artist I really like in Raleigh named Keith Norval.  His paintings are just so goofy and fill me with joy, like this one  called "Waffle time".  "Waffle time" always cracks me up when I see it at night, especially since I love eating breakfast for dinner.  (I used to request breakfast for seminar dinner when I was a visiting speaker.)  I have a different painting of his in my apartment and whenever I look at it I giggle a bit and feel both joy and wonder.  Juanita Pinzón-Caicedo also has a great painting of his too.   Math can fill me with these warm feelings too, in a variety of settings.  Often it's the obvious "status" gains - proving a result I'm proud of, getting a grant, being invited to hang with the cool postdocs as a grad student, etc.  But there's a lot of other things that bring meaningful feelings into my math world.  Giving pep talks and career advice to junior folks is extremely fulfilling; late nights ...

This is a series of posts outlining a proof of the surgery exact triangle in Floer homology. One of the most important features of Floer homology theories for 3-manifolds is the surgery exact triangle.  Let me state this loosely for an arbitrary Floer homology theory $F$.  Suppose that $K$ is a knot in a 3-manifold $Y$ with framing $\lambda$.  Then, there is an exact triangle  $$ \ldots \to F(Y) \to F(Y_\lambda(K)) \to F(Y_{\lambda + \mu}(K)) \to F(Y) \to \ldots   $$ Here, $Y_{\lambda}(K)$ means surgery on $K$ corresponding to the framing $\lambda$, and $\lambda + \mu$ is the framing on $K$ which wraps one more time meridionally than $\lambda$.  (This exact triangle exists as stated, for example, in Heegaard Floer homology, monopole Floer homology, and framed instanton Floer homology.  There are lots of variants of the surgery exact triangle in a bunch of Floer theories.)  The surgery exact triangle has a bazillion applications i...