Caveat lector: let the reader beware. The resources below are byproducts of my various efforts to learn various things over the years -- often through speaking about them. I certainly can't vouch for their accuracy. Nevertheless, I'm keeping them available here, in case others might find them useful.
Beyond being organized by topic, these are in no particular order.
If you'd like, you can show all abstracts or hide all abstracts.
chromatic homotopy theory
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.
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.
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.
The chromatic tower. These are some 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.
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.
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". This is unfinished (I never fleshed out the flowchart of the proof of the main theorem). [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.
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.
black-and-white homotopy theory
The zen of ∞-categories. This is an expository essay extracted from the introductory chapter of my thesis. [abstract]
abstract: In this expository essay, we provide a broad overview of abstract homotopy theory. In the interest of accessibility to a wide mathematical audience, we center our discussion around the theme of (derived)
functors between abelian categories.
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.
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.
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.
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.
Semidirect products are homotopy quotients. This is a short (and extremely outdated) note in which I tried to sort out a hierarchy of various possible interactions between a group and a category. [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.
not homotopy theory
Category O, Soergel bimodules, and the Kazhdan--Lusztig conjectures. These are slides from a talk given in a learning seminar on Soergel bimodules.
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.
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.