Aaron MazelGee
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 lowdimensional topology, and (ii) algebraic Ktheory, 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 "mayzell jee".
I am an active proponent of double conferences, a new ecofriendly conference format whose purpose is to reduce longdistance travel while still fostering longdistance 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 cohosted 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 writeup 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 cohosted 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 coorganizers) 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 Ktheory: Secondary algebraic Ktheory is a categorification of algebraic Ktheory, which was conceived of as an algebrogeometric analog of elliptic cohomology. In forthcoming joint work with my student Reuben Stern, we give a universal characterization thereof, akin to BlumbergGepnerTabuada's universal characterization of algebraic Ktheory. 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 Ktheory. It uses nothing but the geometry of 1manifolds (via factorization homology) and universal properties (coming from Goodwillie calculus). The project comprises three papers:
 A naive approach to genuine Gspectra and cyclotomic spectra,
 Factorization homology of enriched ∞categories, and
 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 nonmathematical audience (though it almost surely isn't wholly so).
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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 ChernWeil 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 traceofmonodromy 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 genuineequivariant homotopy theory. Slides: original/handout.
 I gave three hours' worth of talks (which were not videorecorded), 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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My PhD thesis is entitled GoerssHopkins obstruction theory via model ∞categories. You can see it here.^{*} You can also see the movie adaptation here.
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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.
I have written the following other research papers (aside from the 14 referenced above (click "show more" to see those related to my thesis)).

A relative LubinTate 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.
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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 Ktheoretic localizations of Morava Etheory corepresent a version of the LubinTate moduli problem in this framework.
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A cubical antipodal theorem. This is the result of an REU I did at Harvey Mudd in the summer after my sophomore year, under Prof. Francis Su and with Kyle E. Kinneberg and Tia Sondjaja.
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abstract: The classical LusternikSchnirelmanBorsuk theorem states that if a dsphere 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 dcube 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 dcube by d sets of ridges, at least one set must contain a pair of antipodal kfaces, and we determine the maximum k for which this must occur, for all dimensions except d=5.
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Maximum volume space quadrilaterals. This is the result of a summer research project I did at Brown in the summer after my freshman year, under Prof. Thomas Banchoff and with Nicholas Haber. It was published by the MAA in the book "Expeditions in Mathematics". If it counts, this gives me an Erdős number of 4. (And if we somehow make a movie adaptation, I'll have a Bacon number of 4 too.)
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,
BenZvi,
Carchedi,
Corwin,
DuhlCoughlin,
Farris,
Gepner,
Hansen,
Jordan,
LiBland,
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[{x_{n}:n≠2^{t}1}]=Z/2[x_{2},x_{4},x_{5},x_{6},x_{8},x_{9},...].
The complex cobordism ring is π_{*}(MU)=Z[{x_{2n}}]=Z[x_{2},x_{4},x_{6},...].
a/s/l?