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.
abstract: Algebraic K-theory is a deep and far-reaching invariant, but it is notoriously difficult to compute. To date, the primary means of understanding K-theory is through its "cyclotomic trace" map K→TC to topological cyclic homology. This map is usually advertised as an analog of the Chern character, but this is something of a misnomer: TC is a further refinement of any flavor of de Rham cohomology (even "topological", i.e. built from THH), though this discrepancy disappears rationally. However, despite the enormous success of so-called "trace methods" in K-theory computations, the algebro-geometric nature of TC has remained mysterious.
In this talk, I will present a new construction of TC that affords a precise interpretation of the cyclotomic trace at the level of derived algebraic geometry, which is based on nothing but universal properties (coming from Goodwillie calculus) and the geometry of 1-manifolds (via factorization homology). This is joint work with David Ayala and Nick Rozenblyum.
Homological integration, Feynman diagrams, and the divergence complex. These are notes from a pair of talks given jointly with Eugene Rabinovich in a learning seminar on factorization algebras.
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]
abstract: We prove that a Quillen adjunction of model categories (of which we do not require functorial factorizations and of which we only require finite bicompleteness) induces a canonical adjunction of underlying quasicategories.
Model ∞-categories I: some pleasant properties of the ∞-category of simplicial spaces. [abstract]
abstract: Both simplicial sets and simplicial spaces are used pervasively in homotopy theory as presentations of spaces, where in both cases we extract the "underlying space" by taking geometric realization. We have a good handle on the category of simplicial sets in this capacity; this is due to the existence of a suitable model structure thereon, which is particularly convenient to work with since it enjoys the technical properties of being proper and of being cofibrantly generated. This paper is devoted to showing that, if one is willing to work ∞-categorically, then one can manipulate simplicial spaces exactly as one manipulates simplicial sets. Precisely, this takes the form of a proper, cofibrantly generated model structure on the ∞-category of simplicial spaces, the definition of which we also introduce here.
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]
abstract: We show that the Rezk classification diagram of a relative category admitting a homotopical version of the two-sided calculus of fractions is a Segal space up to Reedy-fibrant replacement. This generalizes the result of Rezk and Bergner on the classification diagram of a closed model category, as well as the result of Barwick and Kan on the classification diagram of a partial model category.
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]
abstract: Goerss--Hopkins obstruction theory is a tool for obtaining structured ring spectra from algebraic data. It was originally conceived as the main ingredient in the construction of tmf, although it's since become useful in a number of other settings, for instance in setting up a "naive" theory of spectral algebraic geometry and in Rognes's Galois correspondence for commutative ring spectra. In this talk, I'll give some background, explain in broad strokes how the obstruction theory is built, and then indicate how one might go about generalizing it to an arbitrary presentable ∞-category. This last part relies on the notion of a model ∞-category -- that is, of an ∞-category equipped with a "model structure" -- which provides a theory of resolutions internal to ∞-categories and which will hopefully prove to be of independent interest.
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]
abstract: In this talk, we begin by introducing algebraic cobordism in analogy with Real cobordism (much like Real K-theory). We then overview the usual buildup of the chromatic story in the motivic setting, noting which parts are formally identical and which parts get hairier. We conclude by describing the motivic Landweber exact functor theorem. This is of course an exciting development in its own right, but it's in some ways not totally satisfactory, since by construction, all Landweber exact homology theories factor through the cellularization functor on motivic spectra.
The chromatic tower. These are some essentially complete notes from a talk I gave in the MIT Juvitop/pre-Talbot seminar on the chromatic tower. There is a video of this talk here. I never wrote this into the notes, but there I managed to wrap things up with a huge diagram of interrelated spectral sequences. [abstract]
abstract: Much of chromatic homotopy theory organizes around the chromatic tower, a tower of certain Bousfield localizations of a given spectrum; the chromatic convergence theorem asserts that the limit of the tower recovers the original spectrum in many cases. After reviewing this background, we'll discuss the proof of the chromatic convergence theorem and then examine the individual layers of the tower.
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]
abstract: Goerss--Hopkins obstruction theory is a tool for obtaining structured ring spectra from purely algebraic data, originally conceived as the main ingredient in the construction of tmf as an E∞-ring spectrum. However, while the story is extremely beautiful, it is also absurdly intricate. Part of this is because the real mathematical ideas at its core are quite deep, but a good deal of the complexity arises from an overwhelming amount of model-categorical technicalities.
In this talk, I will present a generalization of Goerss--Hopkins obstruction theory for presentable ∞-categories. At this level of abstraction, the entire story becomes...well, perhaps not tautological, but certainly a whole lot clearer. It takes a while to get there, though, and so this will be a two-part talk with a short break in the middle (for which I will provide cookies). This all may sound a bit daunting, but I give you my word as a gentleman and a topologist that you will come out with a much better understanding of Goerss--Hopkins obstruction theory than you did going in. Unless you're Mike Hopkins: then I make no promises.
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]
abstract: We compute the homotopy quotient of the G-action on a category of G-objects and nonequivariant morphisms. We begin with a 1-category; a priori this process may yield a higher category, but this turns out not to be the case. Though we don't explicitly pursue it, the argument generalizes readily to ∞-categories enriched in G-spaces. As a corollary, it follows immediately that when the category is a one-object groupoid, the homotopy quotient constructs the semidirect product for the action of G on the automorphism group.
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 appears in the journal Algebraic & Geometric Topology. [abstract]
abstract: We formulate a theory of punctured affine formal schemes, suitable for certain problems within algebraic topology. As an application, we show that the Morava K-theoretic localizations of Morava E-theory corepresent a version of the Lubin--Tate moduli problem in this framework.
You could've invented tmf. These are my slides from the "distinguished graduate student lecture" that I delivered at USTARS in spring 2013. [abstract]
abstract: The cohomology theory known as topological modular forms was first introduced as the target of a topological lift of the Witten genus, an invariant of String manifolds taking values in modular forms. However, it also arises quite naturally in the search for a "global height-2 cohomology theory", i.e. a higher analog of rational cohomology (at height 0) and complex K-theory (at height 1). In this talk, I'll explain what all this means, show how it fits into the bigger picture of stable homotopy theory, and give a step-by-step account of how you, too, could've invented tmf.
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]
abstract:The theory of model categories was originally introduced by Daniel Quillen, that patron saint of the lush pastures between the metropolis of algebra and the jungle of topology. This framework has since become essential on safari, but it's incredibly useful for all you city slickers too -- as an organizational tool and more. I'll give a bit of topological background, show how algebraists secretly use model categories all the time without even realizing it, and then indicate how one can use "generalized spaces" to study chain complexes in non-abelian categories. I'll discuss André-Quillen cohomology as a particular example, and if there's time I may even say a few words about motives and A1-homotopy theory, crystalline cohomology, or algebraic geometry over F1. Not suitable for children under 13.
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]
abstract: These rather skeletal notes are meant for readers who have some idea of the general story of elliptic cohomology. More than anything, they should probably be used as a roadmap when reading the original Survey itself. They grew out of a desire to completely understand the shape of the proof of the main theorem, and so I've postponed the material on equivariant theories until after it in order to make the route to the main theorem as direct as possible. Preorientations and orientations can seem rather mysterious at first; the original paper carries out two examples in great detail. I've also omitted everything from §5, which although fascinating is already quite sketchy in the first place.
Dieudonné modules and the classification of formal groups. This is from a talk in the xkcd seminar in fall 2011. [abstract]
abstract: In this talk, I'll remind you why topologists care about formal groups, introduce the various algebraic objects at play, and illustrate some rather striking classification results.
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]
abstract: In this talk I'll introduce spectra and show how to reframe a good deal of classical algebraic topology in their language (homology and cohomology, long exact sequences, the integration pairing, cohomology operations, stable homotopy groups). I'll continue on to say a bit about extraordinary cohomology theories too. Once the right machinery is in place, constructing all sorts of products in (co)homology you may never have even known existed (cup product, cap product, cross product (?!), slant products (??!?)) is as easy as falling off a log!
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]
abstract, part 1: In this series of talks I'll survey Bott's original proof of his celebrated periodicity theorem. I'll begin by building up the requisite machinery. In this first talk I'll give a rapid overview of Morse theory, and I'll also cover a bit of Riemannian geometry that we'll need in order to analyze the path space of a manifold.
abstract, part 2: In this talk I'll illustrate how we can generalize the framework of Morse theory to obtain a CW-complex which is homotopy equivalent to the path space of a given manifold. This can be made quite explicit for symmetric spaces, which include compact Lie groups as special cases.
abstract, part 3: Last semester, we laid the groundwork for Bott's original proof of his celebrated periodicity theorem. We now return to finish off the story. I'll start by reviewing the Morse theory of path spaces of manifolds; this can be made quite explicit for symmetric spaces, which include compact Lie groups as special cases. Then, we'll use what we've learned to run through Bott's incredibly satisfying proof that Ω2U ~ U (where U = colimnU(n) is the infinite unitary group).
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]
abstract: Higher cohomology operations are an important refinement of the usual "primary" cohomology operations. Roughly speaking, they encode relations between primary cohomology operations. One neat example is the Massey triple product, which (via Poincare duality) can detect the Borromean rings: three circles which are all linked even though pairwise they are unlinked. Higher cohomology operations determine the higher differentials in the spectral sequence for [K,X] from last week's talk, they control composition in the stable homotopy groups of spheres, and they play a crucial role in the Adams spectral sequence.
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]
abstract, part 1: Cohomology is a more powerful invariant than homology. One major reason is that the cohomology groups of a space can be made into a graded ring using the cup product, and induced maps on cohomology must then be ring homomorphisms. However, there is even more structure around. In a series of talks, we'll explore one very important such structure, the action of cohomology operations known as "Steenrod squares". In this first talk, we'll introduce the Steenrod squares, discuss their basic properties, and deduce as a nice little consequence that if the sphere Sn-1 is parallelizable then n=2k. In the longer run, we'll use the Steenrod algebra to compute some homotopy groups of spheres; this is a central and notoriously difficult problem in algebraic topology. The end goal will be to present the Adams spectral sequence, a vital tool which (roughly) computes the homotopy classes of maps from a space X to a space Y.
abstract, part 2: We'll see a number of important concepts (Bockstein homomorphisms, fibrations, Serre's spectral sequence, the transgression, cohomology of K(π,n)'s) that will be lemmatic in our calculation of the homotopy groups of spheres.
abstract, part 3: We'll calculate the stable homotopy groups πn+k(Sn) for 0≤k≤7, n»0, and then we'll calculate either π3+k(S3) or π4+k(S4) for 0≤k≤3 and leave the other as homework.
abstract: The classical Lusternik-Schnirelman-Borsuk theorem states that if a d-sphere is covered by d+1 closed sets, then at least one of the sets must contain a pair of antipodal points. In this paper, we prove a combinatorial version of this theorem for hypercubes. It is not hard to show that for any cover of the facets of a d-cube by d sets of facets, at least one such set contains a pair of antipodal ridges. However, we show that for any cover of the ridges of a d-cube by d sets of ridges, at least one set must contain a pair of antipodal k-faces, and we determine the maximum k for which this must occur, for all dimensions except d=5.