Here are some things I've written and whatnot, roughly in reverse chronological order. If you'd like you can show all abstracts or hide all abstracts.

For the full complement of model ∞-categories papers, please see my home page. I plan to reorganize this page so as to group related writings together (rather than keeping everything chronological), at which point I will add links to those here as well.

The geometry of the cyclotomic trace. These are my slides from various talks I've given on this joint work with David Ayala and Nick Rozenblyum (updated from the version that appears in the talk video). They only cover the introduction; I've been doing the main part of the talk (in which enriched factorization homology and the cyclotomic structure on THH are actually constructed) on the board. [abstract]

Quillen adjunctions induce adjunctions of quasicategories. This fixes a rather embarrassing gap in the literature (imho), and moreover the proof will carry over directly to model ∞-categories. (The key observation is that it's far easier to corepresent operations such as extracting an "∞-category of zigzags" between two objects of a model ∞-category than it is to actually write down simplex-by-simplex coherences. This is achieved by the "model diagrams" introduced here.) It appears here in the New York Journal of Mathematics. [abstract]

Model ∞-categories I: some pleasant properties of the ∞-category of simplicial spaces. [abstract]

From fractions to complete Segal spaces. This is a joint paper with Zhen Lin Low, which has to do with 1-categories but came out of conversations relating to the foundations of model ∞-categories. It will appear in the journal Homology, Homotopy and Applications. (For the record, I would have preferred an Oxford comma in there, but I suppose it's not my prerogative to change their name.) [abstract]

Every love story is a GHOsT story: Goerss--Hopkins obstruction theory for ∞-categories [short version]. These are my slides from my talk at the 2014 Young Topologists Meeting in Copenhagen. As this talk was only 30 minutes instead of 2 hours, these slides are a lot shorter than the ones linked below. I think these provide a much better and cleaner introduction to the theory, and as a bonus these slides also contain some new and (to my mind) compelling results about model ∞-categories. [abstract]

Algebraic cobordism, algebraic orientations, and motivic Landweber exactness. These are the handwritten notes from my (post-Talbot) talk in the MSRI pre-Talbot seminar of spring 2014. [abstract]

Every love story is a GHOsT story: Goerss--Hopkins obstruction theory for ∞-categories [long version]. These are my slides from my talk in the Harvard Thursday seminar at the end of the fall 2013 semester. As I explained the Blanc--Dwyer--Goerss obstruction theory I drew an accompanying diagram, a dramatic reenactment of which you can find here. Moreover, there were a lot of things that I wanted to say that I didn't actually put into the slides themselves (which as you can see are nevertheless still quite overloaded), and many of those things are collected here. [abstract]

Semidirect products are homotopy quotients. This is a short note concerning a hierarchy of various possible interactions between a group and a category. This constitutes a small step in understanding the test case of G-equivariant homotopy theory for the purposes of developing a general theory of "genuine ∞-operads". [abstract]

A Relative Lubin--Tate Theorem via Meromorphic Formal Geometry. This is a joint paper with Eric Peterson and Nat Stapleton, written over the course of my and Eric's stay at MIT in 2013. It will appear in the journal Algebraic & Geometric Topology. [abstract]

You could've invented tmf. These are my slides from the "distinguished graduate student lecture" that I delivered at USTARS in spring 2013. [abstract]

Model categories for algebraists, or: What's really going on with injective and projective resolutions, anyways?. This is from a talk in Berkeley's toolbox seminar in fall 2012. [abstract]

Construction of TMF (or see the notes in context, pp. 16-27). This is from the Uni-Bonn student doctoral seminar in fall 2012 on TMF. The first section ("You could've invented tmf") puts forward a perspective which I think is quite natural, but which I haven't seen it anywhere in the literature (at least not this explicitly). Of course, certainly it's been familiar to the architects of tmf since the beginning.

p-adic modular forms and Dieudonné crystals in stable homotopy theory. This is a poster for GAeL XX. You can see that I was rather pressed for space. The bibliography is here.

Homotopy (co)limits and n-excisive functors (or see the notes in context, pp. 8-20). This is from the Uni-Bonn Arbeitsgemeinshaft in spring 2012 on Goodwillie calculus.

π*LE(1)S for p≠2. This is a short note detailing the computation of the homotopy groups of the E(1)-local sphere at odd primes, transcribed from a conversation with Justin Noel (although all mistakes are certainly my own).

A Survey of Lurie's "A Survey of Elliptic Cohomology". UNFINISHED; mainly I need to finish the flowchart of the main theorem. This is meant to be a companion, not an independent document. Read Lurie's survey for the first bit that I skip over. [abstract]

Dieudonné modules and the classification of formal groups. This is from a talk in the xkcd seminar in fall 2011. [abstract]

Introduction to supermanifolds. This is from a talk I gave in the 80s math for 80s babies (a/k/a Witten in the 80s) seminar. These notes were texed by Theo Johnson-Freyd. Someday I may add a bit that I skipped during the actual talk on Harish-Chandra pairs.

Fibered categories (or see the notes in context, pp. 6-10). This is from a talk in Martin Olsson's seminar on stacks in fall 2011.

The Ballad of the Gallant Hero and the Malicious Adversary: a worksheet on epsilon-delta proofs. This is from teaching math 1A in fall 2011.

An introduction to spectra. This is from a talk I gave in the GRASP (Geometry, Representation theory, And Some Physics) seminar in spring 2011. [abstract]

From Morse theory to Bott periodicity. This is from a sequence of talks in the xkcd seminar, two in spring 2011 and one in fall 2011. [abstracts]

Higher cohomology operations. This is from a talk in the xkcd seminar in fall 2010. The nature of the material makes it nearly impossible to tex up in as clear (if not clean) a way as handwriting with colored pens, so this is a 7.87MB scan. Consider yourself warned. [abstract]

The Steenrod algebra and its applications, part 1 and part 2. These are from a sequence of three talks in the xkcd seminar in fall 2010, but notes from the third would've been lifted directly from Chapter 12 of Mosher and Tangora (on computing homotopy groups of spheres), so I didn't bother. Also, here's a table of the first few monomials in the Steenrod algebra organized by degree and excess. An Adem relations calculator is available here. [abstracts]

A cubical antipodal theorem. This is the result of an REU I did at Harvey Mudd in the summer after my sophomore year, under Prof. Francis Su and with Kyle E. Kinneberg and Tia Sondjaja. [abstract]

Maximum volume space quadrilaterals. This is the result of a summer research project I did at Brown in the summer after my freshman year, under Prof. Thomas Banchoff and with Nicholas Haber. It was published by the MAA in the book "Expeditions in Mathematics". If it counts, this gives me an Erdős number of 4. (And if we somehow make a movie adaptation, I'll have a Bacon number of 4 too.)