Some thoughts on mathematics

We are deep into summer conference season, so let's talk about them.    Often times it’s hard to figure out how to best spend your energy at a conference.  Senior person X might be giving a talk and so you don’t want to be seen missing it.  But maybe she gives terrible talks or you’ve seen it 4 times before and it’s never made sense?  Perhaps if you could understand it, you'd have have the key result you were missing to finish your thesis result.  Or you’re worried she’ll notice you’re not there and is going to think less of you!  On the other hand, you just saw hotshot postdoc Y skipping out on the talk and what if you could get an hour to talk with them?  You’re also almost done with a paper and your collaborator is there – it could be the best time to just knock out the last details while you’re in the same space as him.  You could just be really damn tired from the last 3 days of the conference and going to one more talk is going to drain...

This is a short series on the Smith Conjecture and Smith theory in general   Equivariant topology is always in style, so I figure I should have a blog post about it.  In low-dimensional topology, we're always taking covers (branched or unbranched).  For example, many problems in knot theory are approached by studying the topology of the branched double cover.  (For example, the branched double cover of an unknotting number one knot is Dehn surgery on a knot by something called the "Montesinos trick", and so unknotting numbers get studied in this way.)  So, how can you study group actions on a space: * What is the fixed point set? * What is the quotient manifold? Roughly, if you have a smooth action of a finite group $X^G$ on a smooth manifold $X$, then the fixed point set $X^G$ will be a nice smooth submanifold.  If it is a $p$-group action, where $p$ is prime, then we can really understand the fixed point set well by what is referred to as Smith Theor...

Here's one of my favorite basic algebra/topology things.  You can read about this in Brown's Cohomology of Groups book.   A short exact sequence of groups $$0 \to \mathbb{Z} \to H \to G \to 1$$ is called a central $\mathbb{Z}$-extension of $G$ if the $\mathbb{Z}$ subgroup of $H$ sits in the center of $H$.  For example, $$0 \to \mathbb{Z} \to G \oplus \mathbb{Z} \to G \to 1$$ is the trivial central $\mathbb{Z}$-extension of $G$ and $$0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/2 \to 0$$ is a non-trivial extension, where the first map is multiplication by 2.   Are there other extensions of $\mathbb{Z}/2$?  How do you classify central $\mathbb{Z}$-extensions?  Given an extension and a homomorphism $f: G' \to G$, can we lift $f$ to $H$?  These seem hard to answer, but actually they are easy!!!  And we can topologize this problem.   The one thing we need is group cohomology.  Lightning crash course for this post: if $G...

  This post is mostly a short complaint.  Hopefully more math coming soon now that the semester is over.   Low-dimensional topology can be hard, especially in dimension 4.  As a result, pretty much every paper is of a "counterexample" structure (e.g. these manifolds are not diffeomorphic, this knot cannot bound a disk, etc).   Such papers are made stronger (i.e. get into better journals) if you can give an infinite family of counterexamples to the problem.  Sometimes, I really dislike this approach to papers (even though I am guilty of doing it because I still play the game).  Let me explain just a couple feelings about it.   - Usually there's no reason that this infinite family comprises all possible counterexamples and probably does not really narrow down the possibilities more than having one example, so it is basically answering the same question.  (If it was expected this was the comprehensive list of counterexamples, then...

Here's a cool fact due to Hatcher and W. Thurston ( article here ): 2-bridge knots are small .  I never realized the proof was so simple (and clever), so I wanted to write about it.  I'll briefly explain what small  means and then sketch the proof (which is actually fairly short).   Remark:  They prove something much more general.  They are characterizing incompressible surfaces in the complement of 2-bridge knots, but I won't discuss that here.    Background:  An embedded surface in a three-manifold is incompressible if the inclusion map induces an injection on $\pi_1$.  For example, the boundary of a handlebody is not incompressible but $S^1 \times \Sigma_g$ has many incompressible surfaces: $pt \times \Sigma_g$ or $S^1 \times \gamma$ for $\gamma$ non-trivial in $\pi_1(\Sigma_g)$.  A neat fact is that a knot in $S^3$ is non-trivial if and only if the boundary of the exterior is incompressible.  A knot is small  ...

Everyone seems to be talking about quantum computing, so this semester I might try to learn a little about it, which means you might be subjected to it too.  (Mathematics overcompensates those who follows trends, so why not try to benefit.) Let me try to describe the basic mathematical ideas of how quantum computing works, why it can be good, and why it can be difficult.  I'm going to focus purely on the mathematics - I won't touch on any physics or engineering aspects, which add their own layers of complexity.  But there's a whole world of theoretical quantum computing out there, and it connects with braids, TQFT, and lots of other parts of mathematics! Basic idea:   In classical computers, we might study a bit: it takes the value 0 or 1.  In quantum computers, we consider qubits (quantum bits).  These can kind of be 0 and 1 at the same time.  More precisely, we think of a qubit as $z \mathbf{0} + w \mathbf{1}$ where $z, w \in \mathbb{C}$ and ...

Warning: This short post touches on some heavier topics like suicide and relationship violence.  If you are having a mental health emergency, please reach out to a mental health professional, call 911, or a suicide / crisis hotline (in the US, the canonical number is 9-8-8) .  This post is a bare bones version of what I wish I could write here, but I'd rather get something out and maybe revisit topics in more detail later. Math is weird and has weird people in it.  I remember a close friend in college told me he was a math major because he liked the people - I agree (usually).  Mathematical culture seems to really emphasize being accepting of the weirdos, purporting that a love of mathematics is what matters.  Barefoot at a conference - not even an eyeroll.  (I had my barefoot phase in grad school too.)  Blue hair, you do you!  I like having the space to be weird, but of course, how welcome you are treated in the community can depend on race, gend...