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

I am currently a Sherman Fairchild Instructor in Mathematics at Caltech. I was recently a research member at MSRI for the spring 2020 program on Higher categories and categorification. Before that, I was a Research Assistant Professor at USC.

My research centers around factorization homology, especially as it relates to (i) quantum invariants in low-dimensional topology, and (ii) algebraic K-theory, elliptic cohomology, and chromatic homotopy theory.

My email address is etale.site@aaron, except that you need to swap what comes before and after the "@" symbol.

The first double conference was held in August 2018, entitled Higher algebra and mathematical physics and co-hosted by the Perimeter Institute for Mathematical Physics (Waterloo, Canada) and the Max Planck Institute for Mathematics (Bonn, Germany); you can see the conference websites here and here. You can read a write-up of our experience organizing this event on page 32 of this issue of the London Mathematical Society newsletter.

The second double conference was held in June 2020, entitled Geometric representation theory and co-hosted by the same two institutions; you can see the conference websites here and here. However, due to the global pandemic, it was moved entirely online. Ben Webster (one of the co-organizers) wrote a succinct summary of the lessons we learned regarding online conferences, which you can read as a tweetstorm here.

A third double conference is currently in the works, to be held in summer 2021 and to be entitled Homotopy theory with applications to arithmetic and geometry.

If you might be interested in running a double conference, please feel free to get in touch with me.

update (Oct. '19): We came to understand a key idea in #1 better, and split it out into the new paper Stratified noncommutative geometry. The former will be modified accordingly. [This is substantially broader in scope than "the geometry of the cyclotomic trace"; the foregoing text will eventually be modified accordingly as well.] Here is a video of a talk about this material (slides here).

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.
Here is a video of a talk which focuses on the construction of factorization homology of enriched (∞,1)-categories. Slides here.

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In my thesis, I 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. Using this, I generalize Goerss--Hopkins obstruction theory to an arbitrary (presentably symmetric monoidal stable) ∞-category. As a sample application, I use this generalized obstruction theory to construct E_∞ structures on the motivic Morava E-theory spectra; 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).

The first part of my thesis (regarding model ∞-categories) is assembled from the following papers.

In turn, these are supported by the following supplementary papers (which do not appear in my thesis).

A user's guide to co/cartesian fibrations
From fractions to complete Segal spaces, joint with Zhen Lin Low
Quillen adjunctions induce adjunctions of quasicategories

The second part of my thesis is assembled from the following papers.

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^*Last updated 4/8/2019. I am happy to receive any comments, errata, typos, etc.

I have also split out the first section of the introductory chapter of my thesis 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.

Here is a diagram from a class I taught, which attempts to summarize the relationship between relative categories, model categories, quasicategories, and ∞-categories.

The unoriented cobordism ring is π_{_*}(MO)=Z/2[{x_n:n≠2^t-1}]=Z/2[x₂,x₄,x₅,x₆,x₈,x₉,...].
The complex cobordism ring is π_{_*}(MU)=Z[{x_2n}]=Z[x₂,x₄,x₆,...].
a/s/l?