Its been a while since I’ve written a blog post here. I wanted to give a little briefer on some updates in my life that I’m quite excited about.

First off, my paper Hypercovers and Differential Geometry, joint with Cheyne Glass, has finally hit the arXiv. I say finally, because I’ve been thinking about hypercovers and the issues this paper addresses since I was just starting out as a graduate student. It revolves around a technical issue that crops up when dealing with model structures on simplicial presheaves. In higher differential geometry–which my thesis Higher Diffeology Theory was about, and which my coauthor Cheyne, his advisor Thomas and my advisor Mahmoud have written a bunch of papers about–we want to study manifolds using sheaves of spaces, i.e. simplicial presheaves. We usually deal with the site \((\mathbf{Man}, j_{\text{open}})\) of finite-dimensional smooth manifolds with open covers, or \((\mathbf{Cart}, j_{\text{good}})\), the site of cartesian spaces (spaces diffeomorphic to some \(\mathbb{R}^n\)) with differentiably good open covers. For technical reasons, the latter site is much preferred. So, as is done in my thesis, a big part of higher differential geometry is extracting cocycles from simplicial presheaves on manifolds. Given a site \((\mathcal{C}, j)\), there are two important model structures one can put on the category \(\mathbf{sPre}(\mathcal{C}, j)\) of simplicial presheaves, the Cech projective model structure \(\check{\mathbb{H}}(\mathcal{C}, j)\) and the local projective model structure \(\hat{\mathbb{H}}(\mathcal{C}, j)\). The former has really nice fibrant objects, called \(\infty\)-stacks, and the latter has really nice weak equivalences, called the local weak equivalences.

The first presents the infinity category of infinity sheaves on \((\mathcal{C}, j)\) and the latter presents the hypercompletion of this infinity category. On certain sites, these two infinity categories are equivalent. We call sites with this property hypercomplete. Its been known for a while now that \(\mathbf{Man}\) and \(\mathbf{Cart}\) are hypercomplete, using some material from the latter parts of Lurie’s Higher Topos Theory. However, the more useful thing is to ask when are the Cech and local model categories equal. This gives one the best of both worlds: local weak equivalences for weak equivalences and \(\infty\)-stacks for fibrant objects. This paper proves this property for a couple of sites including \(\mathbf{Man}\) and \(\mathbf{Cart}\).

This project was close to my heart, as understanding the issues surrounding it led me to more deeply understand sites, leading to my notes Coverages and Grothendieck Toposes. I think of this paper as the final sort of piece of my extended PhD thesis, which would basically consist of Higher Diffeology Theory, Coverages and Grothendieck Toposes and Hypercovers in Differential Geometry. There is still so much to do, especially in regards to the coverages notes, but I don’t really have much energy for it these days.

Instead, I’m really excited about applying abstract math, like category theory, homotopy theory, etc. to discrete mathematical problems. I’ve been learning about a whole ton of really fascinating stuff, like graph homotopy theory, the use of toric varieties in combinatorics, and intersection homology. I really love the mix of abstract methods to solve concrete problems. In this vein, my paper Thomason-Type Model Structures on Simplicial Complexes and Graphs has just been published by Applied Categorical Structures! In it, I prove that you can transfer the classical model structure on simplicial sets to simplicial complexes, reflexive graphs and loop graphs, leading to a whole sequence of Quillen equivalences, using a similar method that Thomason used for his model structure on \(\mathbf{Cat}\), and I elaborate on some properties of these model categories. This provides some more details on Matsushita’s construction of a model structure on loop graphs.

Speaking of graph homotopy and homology, I made some updates to my list of papers on the subject at the discretepapers tab. There’s an endless list of fascinating papers to read on this subject, but hopefully soon I’ll have most of the big ones listed there.

On the teaching side, I’m teaching a really cool course: introduction to category theory at CUNY CityTech. This is the first time a course like this has been taught to undergraduates there, and I’m having loads of fun. I’m aiming to first introduce some of the basic discrete mathematical structures: sets, posets, monoids, graphs and simplicial complexes, and then introduce the basic ideas of category theory through these examples. I’ll try and report back my thoughts on this later in the semester.

I’ve also become co-organizer of the NYC Category Theory seminar, which has also changed from a 7pm time to a 2pm time on Wednesdays, which is way more convenient. We’ve got a really great lineup this semester, which you can check out here.

I’ll be giving a talk at a conference Homotopy and Homology in Graphs in Switzerland at EPFL in September about my Thomason-Type model structures paper. I’m really excited about this, shoot me an email if you’ll be attending as well!

I’ll probably go to another conference during the summer, but there are so many good choices, I’m not sure yet which I’ll choose.

I’m super excited for this year, but I’m not going to lie, entering a new field feels really daunting. I’ve been thinking about higher differential geometry for my whole PhD. Now I have to retrain myself to think more like a combinatorialist. Its probably a bad idea to switch fields this early in one’s math career, but I’m confident that it’ll be okay, the upside being that this new area (abstract math applied to discrete math) has loads of fascinating things going on. In one direction I’m in awe of the fast-growing theory of algebraic topology being applied to graphs. Check out my discretepapers tab if you want to see a bunch of cool papers, and in another, I’m working with Ben Bumpus on some more stuff related to category theory, graph theory and algorithms. Hopefully more on that soon! I’ve been learning a lot of graph theory, and especially about different graph classes. These graph classes are really fascinating, and studying them makes me feel like there could be really interesting invariants one could extract from them. Like I want to treat a whole graph class of a certain kind as a single mathematical object (maybe like a spectrum in homotopy theory). Anyway, this post is just some rambling, maybe I’ll ramble some more soon.