## Pullbacks and Intersections

### June 17, 2021

Sometimes a pullback square is an intersection! Here’s a short post about something cute that came up in my research this week. Aside from NCNGT, which has been very lovely, my math this week has been carefully walking through a small side proposition in a paper I’m working on with Ty Ghaswala. The idea is that graphs of groups come with a notion of a covering space due to Bass, and étale groupoids come with their own notion of covering spaces (for us, we’re following the definition in Bridson–Haefliger), and given that you can construct an étale groupoid associated to any graph of groups, I wanted to reconcile the two notions.

## Nielsen Realization

### June 12, 2021

I’m giving a talk in Nearly Carbon Neutral Geometric Topology about Nielsen realization for infinite-type surfaces. The format for the conference is that everyone records and uploads their talks beforehand, and then the conference format allows for discussion and comments via Discord. The purpose of this post is to share and talk a little bit about my talk, which is below. The Nielsen realization problem was posed by Nielsen in 1932.

## A Mathematical Postcard

### June 9, 2021

I submitted a “mathematical postcard” (see below) to the Nearly Carbon Neutral Geometric Topology conference. The purpose of this post is to share the postcard and offer a little more context and explanation for it. I’m also giving a talk in said conference, but I’ll talk about that in a separate post. Here is the postcard: Let me say a little more about the space we’re talking about. Let $F = A_1*\dotsb*A_n*F_k$ be a free product of the groups $A_1,\dotsc,A_n$ with a free group of rank $k$ .

## Notes on Orbifolds

### March 14, 2021

The purpose of this post is to share some talk notes I made on orbifolds and graphs of groups as étale groupoids. The talk comes out of joint work in progress with Tyrone Ghaswala. It starts with us realizing that some representations of the braid group into the automorphism group of a free group that each of us knew how to obtain by different methods were really the same representation. We thought that there ought to be some way to reconcile these two representations topologically by working with the mapping class group of certain 2-dimensional orbifolds.

## The Train Track Algorithm for Graphs of Groups

### December 21, 2020

Train track maps, which also come in “relative” and other flavors, are particularly nice homotopy equivalences of graphs used as tools to reason about outer automorphisms of free groups. I wrote a paper extending train track maps (as well as relative train track maps and “CTs”) to graphs of groups (with trivial edge groups for CTs). In this post I want to talk a little about how to think about train track maps and (“tame” in some sense) homotopy equivalences of graphs of groups more generally.

## The Axiom of Choice Is False for Manifolds

### August 2, 2020

This is a post about one of those things that is really a case of category theory cheekily playing with unhooking a signifier from what it typically signifies. Namely, one of the (many) tricky foundational things about working with orbifolds goes by the name “the axiom of choice is false for the category of smooth manifolds.” One way to misinterpret the statement is to take it literally: as undergraduate math majors learn, the axiom of choice says that given an infinite collection of sets, you can always choose exactly one element of each set, even if the collection is infinite.

### July 26, 2020

Given a topological space $X$ and a continuous map $\pi\colon X \to Y$ , there is an adjunction between the categories of sheaves on $X$ and sheaves on $Y$ . The adjunction is somewhat mysterious to me as I begin to write this, so the hope is that by writing about it, I’ll begin to understand. The functors in question are the pushforward and inverse image functors, which arise naturally in algebraic geometry (so I am led to believe).

## Two Approaches to the Cotangent Space

### July 10, 2020

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.

## Orbifolds Day One

### July 5, 2020

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.

## Category Theory Day One

### February 16, 2020

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.