There is a “sheaf-theoretic” definition of something like a cotangent space at a point for a locally-ringed space. In the case of the sheaf of differentiable functions on a smooth manifold, the two definitions—sheaf-theoretic and the usual one in differential geometry—agree. What’s more, modulo the appropriate background technology in both subjects, the proof is beautiful and simple—I feel like shouting it from the rooftops at the moment, so this blog post will try and give a little exposition.

I wrote some notes introducing orbifolds from Haefliger’s perspective in Metric Spaces of Non-positive Curvature. There is some surprisingly heavy machinery that goes into it. In this blog post I want to talk a little about it at a high level. An orbifold of dimension $n$ is a Hausdorff space locally modeled on $\mathbb{R}^n$ modulo the action of a finite group. This is a simple enough idea on its face: a useful perspective on the fundamental group of a manifold is the passage between its action on the universal cover and the calculus of loops in the base space.

My office-mate asked me to give a “second lecture” in category theory in Eminar, the Tufts graduate student seminar. I gave a first lecture last semester, but I thought it might be nice to have something to point people who missed it to, in order to stick to my assignment and begin at the Yoneda lemma. So, here is yet another rendition of the first hour of a category theory class.

Today I posted my first paper, Train Tracks, Orbigraphs and CAT(0) Free-by-cyclic Groups to the ArXiv! Here is a description of the paper and the idea behind the main application. Orbigraphs One of my main goals currently is to better understand outer automorphisms of free products of finite groups. I became interested because I like Coxeter groups, and I was surprised to hear that we knew so little about their automorphisms.

I’m not sure what this post set out to be. It ended up a digression on set theory vs. category theory. In modern mathematics, there are two notions that might be truly called foundational to all the work we do. These are set theory and category theory. Both theories are living, evolving things with their specialists, constructions and interesting problems, just like any other field of math, but the core ideas are familiar and useful to nearly every mathematician.

You know the sensation where a familiar word becomes suddenly completely strange for no apparent reason? Like suddenly the scales fall away from your eyes and you see how utterly weird it is that a concept has a particular name? I’ve been having that feeling with the word graph as I use it in math. As a kid, you learn how to graph simple functions in the sense of being able to sketch them on a sheet of, say, graph paper.

Earlier this week, Marc Weidenbaum of disquiet.com wrote a paean to the blog that I found really inspiring. “Self-publishing,” he writes, “is the heart of the healthy internet.” So hello. Having been given the push I needed, I decided to move my academic website to its own URL and use it as an excuse to start blogging. Since much of the rest of this website is given over to my career as a mathematician, it’s reasonable to expect that I will talk about math here, and I expect to get technical with it when I want to.