Some thoughts on mathematics

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...

Sometimes I think about math which is not pure geometry/topology, so I want to discuss some of those areas in this blog as well.  This post was inspired by a recent talk of Moon Duchin in MD4SG and some discussions I had with Jim Allen and Manon Revel.  This is an oversimplification to get the ideas across - there are a lot of important ideas not present in this post.  If you want to learn more, check out Moon's group's  website , which has lots of nice visuals and interactive explanations.  Let me start with the background before getting into the math.  A major source of political conflict comes from gerrymandering .  If you're not familiar with this term, it's roughly the idea of restructuring voting districts to create some kind of political advantage.  I'll explain this a little further through the lens of the US system.  For simplicity, let's assume there are two political parties A and B in a state, which will be divided into vo...

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...

  In this post, we describe how ASD connections on four-manifolds relate to Chern-Simons theory on three-manifolds. Suppose that we have a smooth four-dimensional cobordism $W : Y_1 \to Y_2$ between homology three-spheres and consider the trivial $SU(2)$ bundle on $W$.  Just like for closed manifolds, we can count ASD connections on $W$.  The very rough idea is that an ASD connection restricts to a flat connection on the $Y_i$, and we can use it to relate the Chern-Simons invariants of $Y_1$ and $Y_2$.  (I lied a little, but I explain the issue in a technical point below.)  At the end of this post, we'll show that there is no simply-connected homology cobordism from the Poincare sphere to itself.  Get pumped! Let's be a little more precise with our gauge theory now.  For a connection $A$ on $W$, we can measure its  energy:    $$ \mathcal{E}(A) = \int_W F_A \wedge F_A. $$  If you don't like the formula, there are two relevant fa...

This popped in my head, so let me make this post short. I am anti-colloquium.   I never remember the content of the non-geometry/topology colloquia.  Not even general guiding principles.  The only exception is I remember that a lot of people care about modular forms, whatever those are.  Plenty of colloquium talks are poorly executed by the speaker, but many I actually enjoy and understand some amount of until I fall asleep.  But then it's immediately gone when I leave the room.  I personally don't think colloquium enhances my research or teaching; I don't think it helps my mentees either, unless it's in geometry/topology or maybe demonstrating how to give a good (or bad) talk.     I recognize that there's other value to the colloquium.  As one example, there aren't many departmental events that bring together all students and faculty, especially if tea is tied entirely to the colloquium.  It is important to keep the department from...