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Surgery exact triangles (side quest)

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In a previous post I started to talk about how to prove the surgery exact triangle in Floer homology.  I hope to return to this series, but I thought I would give a simple application of the exact triangle to present an alternate proof of a recent result of Hom and Wan, where they show that the -1-traces on the knot $5_2$ and the pretzel knot $P(3,-3,-8)$ are non-diffeomorphic, even though these four-manifolds are homeomorphic.  (In other words, these are absolutely exotic four-manifolds.)  This is a very special case of their more interesting and general result relating the Heegaard Floer $\nu$-invariant to four-dimensional traces.  I thought I would instead present a goofy ad hoc argument, not for the purpose of shedding light on their work, but rather to give some more perspective on how the exact triangle can be useful.  (What is done below is secretly related to their more general work on the $\nu$-invariant, but not in an obvious way.) We will start wit...
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Front of the crowd

If you don't know the legendary punk/ska band Operation Ivy, you probably aren't missing out on anything.  However, if you do know them, and the only album they put out back in 1989, then you may appreciate knowing that their legacy has been preserved by the combination of Catbite and Laura Jane Grace playing their album live  here .  However, this post - which I promise is relevant to mathematics - is not about the glory of ska music, nor the important role that Laura Jane Grace played being one of the first trans punk musicians to publicly come out. This post is actually about a random person in the audience in that video.  (Disclaimer: I don't personally know this person, they just happen to be a good metaphor, so my apologies to them for being a focal point of criticism in this post.) Let me give some context so you don't have to watch the video.  At many punk shows, the cultural norm is to have some active engagement with the music - singing along...
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At the merch table

(Thanks to John Baldwin, Juanita Pinzón-Caicedo, and Steven Sivek for comments and moral support.)   In grad school, the AMS (American Mathematical Society) would have merch tables at conferences where they sold t-shirts.  These shirts were pretty goofy with slogans like "You want the proof, you can't handle the proof".  At the time, I thought I was too cool for that.  But for a brief period, maybe around 2013 or so, the AMS had these long sleeve shirts that I thought were surprisingly stylish.  I thought the AMS was finally "with it".  One shirt I got is apparently cool enough that a hipster ex still wears it (and I'm probably not getting it back).  For the past 10 years or so I've checked the AMS website to see if they've got cool new math gear, and I'm always very disappointed with the shirts.   This week I'm disappointed with the AMS for a more serious reason.  Recently, they put out the following statement around the Navier-Stokes f...
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Waffle time feelings

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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 ...
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Surgery exact triangles (part 1)

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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...
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Advice on advice

Recently I was at a conference for graduate students that featured a panel on collaboration.  This was super cool - I really like the idea of having some professional development panels at a conference, especially ones that I haven't seen at conferences in the past.  There was a lot to take in, but I think there was some really good advice on collaboration.  I had some tangential thoughts from the panel experience I wanted to share.  Warning:  I don't think what's written below may be that helpful or even cohesive, but I'm still going to post it in case it's useful for sparking some conversation.     I asked a graduate student after the collaboration panel what they thought about it.  This student explained to me that they just started research recently - collaboration was not on their mind at all.  Why am I pointing it out?  Often, we feel we need to be put on high alert for all things: make sure to seek out any possible ...
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Contractible four-manifolds that are not Mazur-type (part 2)

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Back for more!  Recall that we are interested in proving the following proposition:  Proposition:  There exists an irreducible, i.e. prime, homology 3-sphere which bounds a contractible 4-manifold and no bounding contractible four-manifold can be built with fewer than five handles.   In the last post, we showed that it suffices to show that there is a prime homology 3-sphere that bounds a contractible 4-manifold that is *not* surgery on a knot in $S^2 \times S^1$.  That was because being surgery on a knot in $S^2 \times S^1$ is equivalent to bounding a Mazur manifold, and those are the only contractible 4-manifolds with a handle decomposition with fewer than five handles.  That's our goal.  If you don't care about a proof of the proposition, here's a surgery picture for an example (and if you do care, use this picture as reference for the constructive proof below):   We'll start the proof by giving an obstruction to a homolo...
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