Aaron Mazel-Gee's Home Page on the World Wide Web!

In paper 16, we noticed a very cute cancellation of two failures. As background, note that the gluing functors for genuine G-spectra are given by Tate constructions (see there for more explanation). Now, over the integers, neither the gluing functors nor the Tate construction is tensored up from the sphere spectrum. However, the gluing functors over the integers are nevertheless given by Tate constructions. (In fact, this is true over any commutative ring spectrum.) This leads to a substantial simplification in the resulting description of derived Mackey functors, which gives a new method of computing equivariant cohomology. As a proof of concept, we give a complete computation of the C_pⁿ-equivariant cohomology of a point (which was previously known only for n≤2).

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Secondary algebraic K-theory is a categorification of algebraic K-theory, which was conceived of as an algebro-geometric analog of elliptic cohomology. We give a universal characterization thereof, akin to Blumberg--Gepner--Tabuada's universal characterization of algebraic K-theory.

Here is a video of a talk about this material (slides here).

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We came to understand a key idea in paper 11 better, which became the germ of this paper.

We added a bunch of really cool material to arxiv v2 of this paper that wasn't in v1. If you liked v1, you'll love v2! (See "comments" on the arxiv v2 page for specifics.)

Here is a video of a talk about stratifications and reconstruction (slides here).

Here are slides from a talk that focuses on reflection.

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We establish a new construction of topological cyclic homology (TC) that affords a precise interpretation at the level of derived algebraic geometry of the cyclotomic trace K → TC from algebraic K-theory. In contrast with the original construction in terms of equivariant homotopy theory, it uses nothing but the geometry of 1-manifolds (via factorization homology) and universal properties (coming from Goodwillie calculus). (This relies on results established in papers 11 and 12 below, but it can be read entirely independently of them.)

Here is a short description of the project which is intended to be readable by a non-mathematical audience (though it almost surely isn't wholly so).

Linked below are some videos and slides from talks that I've given about this material. As for the slides, all are available in their original format as well as in "handout" format (so with far fewer pages). However, beware that because of various little TeX/beamer hacks that I used in making these, the latter are a little screwy in places (e.g. a few words are repeated (in different colors) and some material is cut off from the bottom of some slides). Also, note that the slides are all slightly edited from the versions appearing in the videos.

Here is a video of a talk for a broad geometry & topology audience (assuming no familiarity with algebraic geometry, let alone derived algebraic geometry). It begins by recalling an analogous story regarding the construction of the Chern character in differential geometry via Chern--Weil theory, and concludes with the main (derived algebro-)geometric picture: that TC(X) consists of those functions on the free loopspace LX of the scheme X satisfying certain conditions that are naturally present on trace-of-monodromy functions of vector bundles. Slides: original/handout.
Here is a video of a talk which describes the main geometric picture and then explains the construction of TC via factorization homology and Goodwillie calculus. Slides (just for the first part -- the construction of TC was on the blackboard): original/handout.
Here is a video of a talk which describes an enhancement of the main geometric picture -- a stratified stack that encodes the "cyclotomic" symmetries of LX --, which illuminates the connection with genuine-equivariant homotopy theory. Slides: original/handout.
I gave three hours' worth of talks (which were not video-recorded), discussing essentially everything described above and including more details in a few places. Slides: original/handout.

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Here is a video of a talk about this material. Slides here.

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For the record, I would prefer an Oxford comma in the journal name, but I suppose it's not my prerogative to change it.

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I introduce and develop the theory of model ∞-categories -- that is, of model structures on ∞-categories -- which provides a robust theory of resolutions that is entirely native to the ∞-categorical context (papers 3, 6, 7, 8, 9, and 10). Using this, I generalize Goerss--Hopkins obstruction theory to an arbitrary (presentably symmetric monoidal stable) ∞-category (paper 14). As a sample application, I use this generalized obstruction theory to construct E_∞ structures on the motivic Morava E-theory spectra (paper 15); just as in the classical case, these E_∞ structures turn out to be essentially unique, and their automorphism groups turn out to be essentially discrete (though they will generally be strictly larger than the usual Morava stabilizer group).

I have split out the first section of the introductory chapter into an essay called The zen of ∞-categories, which you can see here. This is an introduction to abstract homotopy theory. In the interest of accessibility to a broad mathematical audience, it is centered around the classical theory of abelian categories, chain complexes, derived categories, and derived functors.

I've given a number of talks about this material, most of which have been entitled Every love story is a GHOsT story: Goerss--Hopkins obstruction theory for ∞-categories (a reference to one of my preferred authors).

Among these, I first shared my work in a two-hour talk at the Harvard Thursday seminar at the end of the fall 2013 semester. The slides for that talk are here. 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.

I also gave a 30-minute talk on this work at the 2014 Young Topologists Meeting in Copenhagen, in which of course I had to get to the point much more quickly. The slides for that talk are here. I think these provide a much better and cleaner introduction to the theory, and as a bonus they also contain some new and (to my mind) compelling results about model ∞-categories.

You can also see a movie adaptation of my thesis here.

I last updated my thesis on 4/8/2019. I am happy to receive any comments, errata, typos, etc.

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