Aaron Mazel-Gee

I am currently a Sherman Fairchild Instructor in Mathematics at Caltech. My office is 285 Linde Hall.

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 work is partially supported by NSF grant DMS-2105031 (Factorization homology and low-dimensional topology).

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".

^† = undergraduate advisee

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in progress:

• Perverse schobers and abelian 3d mirror symmetry, with Benjamin Gammage and Justin Hilburn

  Here is a video of a talk by Justin about this material.

• Factorization homology beta in algebraic geometry, with Quoc Ho

• Khovanov homology and factorization homology, with Leon Liu^† and David Reutter

• Cyclotomic Khovanov spectra, with Michael Willis and Matthew Stoffregen

1. Derived Mackey functors and C_pⁿ-equivariant cohomology, with David Ayala and Nick Rozenblyum, 05/06/2021
   arxiv:2105.02456, 83 pages.
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   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 only known for n≤2).

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2. A universal characterization of noncommutative motives and secondary algebraic K-theory, with Reuben Stern^†, 04/08/2021
   arxiv:2104.04021, 80 pages.
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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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3. Dualizable objects in stratified categories and the 1-dimensional bordism hypothesis for recollements, with Grigory Kondyrev and Jay Shah, 03/29/2021
   arxiv:2103.15785, 61 pages.

4. Stratified noncommutative geometry, with David Ayala and Nick Rozenblyum, 10/31/2019
   arxiv:1910.14602, 190 pages.
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   We came to understand a key idea in paper 11 better, which became the germ of this paper.

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

   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.)

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5. E_∞ automorphisms of motivic Morava E-theories, 01/17/2019
   arxiv:1901.05713, 6 pages.

6. Goerss--Hopkins obstruction theory for ∞-categories, 12/18/2018
   arxiv:1812.07624, 54 pages.

7. The geometry of the cyclotomic trace, with David Ayala and Nick Rozenblyum, 10/17/2017
   arxiv:1710.06409, 48 pages.
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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.

   1. 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.

   2. 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.

   3. 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.

   4. 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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8. Factorization homology of enriched ∞-categories, with David Ayala and Nick Rozenblyum, 10/17/2017
   arxiv:1710.06414, 68 pages.
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   Here is a video of a talk about this material. Slides here.

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9. A naive approach to genuine G-spectra and cyclotomic spectra, with David Ayala and Nick Rozenblyum, 10/17/2017
   arxiv:1710.06416, 84 pages.

10. Model ∞-categories III: the fundamental theorem, 10/16/2015
    New York Journal of Mathematics, 27 (2021), 551-599.
    arxiv:1510.04777, 34 pages.

11. Model ∞-categories II: Quillen adjunctions, 10/15/2015
    New York Journal of Mathematics, 27 (2021), 508-550.
    arxiv:1510.04392, 29 pages.

12. Hammocks and fractions in relative ∞-categories, 10/14/2015
    Journal of Homotopy and Related Structures, 13 (2018), no. 2, 321-383.
    arxiv:1510.03961, 43 pages.

13. On the Grothendieck construction for ∞-categories, 10/13/2015
    Journal of Pure and Applied Algebra, 223 (2019), no. 11, 4602-4651.
    arxiv:1510.03525, 41 pages.

14. The universality of the Rezk nerve, 10/12/2015
    Algebraic & Geometric Topology, 19 (2019) no. 7, 3217-3260.
    arxiv:1510.03150, 26 pages.

15. A user's guide to co/cartesian fibrations, 10/08/2015
    Graduate Journal of Mathematics, 4 (2019), no. 1, 42-53.
    arxiv:1510.02402, 16 pages.

16. Quillen adjunctions induce adjunctions of quasicategories, 01/13/2015
    New York Journal of Mathematics, 22 (2016), 57-93.
    arxiv:1501.03146, 20 pages.

17. Model ∞-categories I: some pleasant properties of the ∞-category of simplicial spaces, 12/29/2014
    arxiv:1412.8411, 66 pages.

18. From fractions to complete Segal spaces, with Zhen Lin Low, 09/29/2014
    Homology, Homotopy and Applications, 17 (2015), no. 1, 321-338.
    arxiv:1409.8192, 21 pages.
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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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19. A relative Lubin--Tate theorem via meromorphic formal geometry, with Eric Peterson and Nathaniel Stapleton, 08/25/2013
    Algebraic & Geometric Topology, 15 (2015) no. 4, 2239-2268.
    arxiv:1308.5435, 18 pages.

0. Goerss--Hopkins obstruction theory via model ∞-categories, 05/13/2016
   545 pages.
   This comprises papers 3, 6, 7, 8, 9, 10, 14, and 15 above, plus an introductory chapter (76 pages).
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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, 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 here.

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

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-1. A cubical antipodal theorem, with Kyle E. Kinneberg, Tia Sondjaja, and Francis Su, 09/02/2009
    arxiv:0909.0471, 15 pages.
    This is the result of an REU I did in the summer after my sophomore year, supervised by Francis Su.

0. Maximum volume space quadrilaterals, with Thomas Banchoff and Nicholas Haber, 08/02/2006
   Expeditions in Mathematics, 2 (2011), 175-198.
   23 pages.
   This is the result of a summer research project I did in the summer after my freshman year, supervised by Thomas Banchoff. 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.)

1. An invitation to higher algebra (in progress, see here for more info)

I passed my qualifying exam on Friday, May 13, 2011. You can see the syllabus here.

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?

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.