Aaron Mazel-Gee

Hello! My name is Aaron. I like to do math. It is my job.

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.

You can see my c.v. here.

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

My last name is pronounced "may-zell jee".

I am an active proponent of double conferences, a new eco-friendly conference format whose purpose is to reduce long-distance travel while still fostering long-distance interactions. Events take place in two different locations connected by live video stream.

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.

A universal characterization of secondary algebraic K-theory: Secondary algebraic K-theory is a categorification of algebraic K-theory, which was conceived of as an algebro-geometric analog of elliptic cohomology. In forthcoming joint work with my student Reuben Stern, 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).
The geometry of the cyclotomic trace: In collaboration with David Ayala and Nick Rozenblyum, I have developed a new construction of topological cyclic homology (TC) that accommodates a precise interpretation at the level of derived algebraic geometry of the cyclotomic trace K → TC from algebraic K-theory. It uses nothing but the geometry of 1-manifolds (via factorization homology) and universal properties (coming from Goodwillie calculus). The project comprises three papers:
  1. A naive approach to genuine G-spectra and cyclotomic spectra,
  2. Factorization homology of enriched ∞-categories, and
  3. The geometry of the cyclotomic trace.
The main results are contained in paper 3, which is intended to be accessible to a relatively broad audience. (In particular, it can be read entirely independently of papers 1 and 2.)

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).

update (Oct. '20): We added a bunch of really cool material to Stratified noncommutative geometry. If you liked arxiv v1, you'll love arxiv v2! (See comments on v2 for specifics.)

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).

[show additional resources]

My PhD thesis is entitled Goerss--Hopkins obstruction theory via model ∞-categories. You can see it here.* You can also see the movie adaptation here. [show more]

*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.

I have written the following other research papers (aside from the 14 referenced above (click "show more" to see those related to my thesis)).
The Adem relations calculator is here -- brought to you, as always, by the wizardry of the kruckmachine.

I passed my qualifying exam on Friday, May 13, 2011.

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

The DavidRoll: Alper, Antieau, Ayala, Ben-Zvi, Carchedi, Corwin, Duhl-Coughlin, Farris, Gepner, Hansen, Jordan, Li-Bland, Nadler, Orman, Penneys, Reutter, Roberts, Spivak, Treumann, White, Yetter.

writing       talks       teaching       conferences       service       PR       xkcd seminar       livetex

The unoriented cobordism ring is π*(MO)=Z/2[{xn:n≠2t-1}]=Z/2[x2,x4,x5,x6,x8,x9,...].
The complex cobordism ring is π*(MU)=Z[{x2n}]=Z[x2,x4,x6,...].