So it’s finals season, and earlier today some of the younger grad students were asking me for help studying for their topology finals. One of their practice problems was to build a cell complex with one 0-cell, two 1-cells, and two 2-cells which has nontrivial $\pi_1$ but trivial $H_1$. In principle, this isn’t very hard to do – I encourage you to think about it for a while to see what kind of…
Today Yesterday in the Representation Theory Seminar at UCR I gave a talk about Factorization Homology and how it lets us compute a “Quantum Character Stack”. This is all based on a great paper, Integrating Quantum Groups Over Surfaces by Ben-Zvi, Brochier, and Jordan, which I’ve been reading and rereading for the last few years. It’s been a while since I’ve written up my thoughts after a talk, so…
In the main post I go over a talk I gave today yesterday explaining how factorization homology relates to (quantum) character stacks. In this post I want to do a sample computation which gives some justification that the main theorem from that talk (which is not mine) actually does work! Recall from that blog post that we can compute the category of sheaves on the character stack…
Yesterday I watched my friend Jialin Wang defend her thesis, and as part of her background section she mentioned that the group $F_2 \times F_2$ is incoherent in the sense that it has a subgroup that’s finitely generated and not finitely presented. I was curious how one might prove something like this, and in the original paper (Stallings’s Coherence of 3-Manifolds Fundamental Groups ) this fact…
I’ve spent the last week at CT2025 , which has just come to a close. It was great getting to see so many old friends and meet so many new ones, and every time I go to a CT I’m reminded of just how much category theory there is in the world, as well as just how much I enjoy all of it! Right before this I was in Antwerp for some Noncommutative Geometry , where I learned a ton and met even more new…
Earlier this week my friend Shane and I took a day and just did a bunch of computations. In the morning we did some differential geometry, where he told me some things about what he’s doing with symplectic lie algebroids . We went to get lunch, and then in the afternoon we did some computations in derived algebraic geometry . I already wrote a blog post on the differential geometry, and now I want…
This is going to be a very classic post, where we’ll chat about a computation my friend Shane did earlier today. His research is largely about symplectic lie algebroids , and recently we’ve been trying to understand the rich connections between poisson geometry, lie algebroids, lie groupoids, and eventually maybe fukaya categories of lie groupoids (following some ideas of Pascaleff ). Shane knows…
While doing a computation with my friend Shane the other day, we realized we needed to explicitly compute a local chart near the identity of $SL_2(\mathbb{R})$. It took us longer than I’d like to admit to figure out how to do this (especially since it’s so geometrically obvious in hindsight), and so I want to write down the process for future grad students looking to just do a computation! If you…
The other day my friend Lucas Salim was asking me some questions about categorical logic and constructive math, and he mentioned he’d never seen a proof that there’s no constructive proof of the intermediate value theorem before. I showed him the usual counterexample, and since my recent blog post about choice was so quick to write I decided to quickly write up a post about this too, since I…
A few days ago I saw a cute question on mse asking about a particularly non-intuitive failing of the axiom of choice. I remember when I was an undergrad talking to a friend of mine about various statements equivalent to choice, and being particularly hung up on the same statement that OP asks about – The product of nonempty sets is nonempty. I understood that there were models where the axiom of…