The Étale Site

The Big Étale Site: Life in the Large (Present & Future)

AI

I'm working at Kerna Labs to make AI that makes mRNA-based medicines. I previously worked at Tilde performing research in mechanistic interpretability -- a field often referred to affectionately as "neuroscience for AI". I've also previously worked in software engineering in various contexts as well.

A few resources generated and collected from my engagement with AI are available here.

writing

As its name indicates, my Substack is dedicated to explorations of love, truth, and beauty. I intend to write about: mathematics; AI; meditation; running; piano; dance; romance and intimacy; Burning Man; emotional processing; et surely alia. More info is available here. Please subscribe if you'd like to receive updates!

math & academia

I obtained my PhD at UC Berkeley, and was most recently employed teaching and researching mathematics at Caltech. I've always been a teacher at heart, and I intend to continue to embody that spirit throughout my life -- particularly through my writing.

My research was incredibly beautiful, but also incredibly abstract: the most one could say about its relevance in the real world is that it is "theoretically applicable to theoretical physics" (specifically to quantum field theory -- the high-energy refinement of quantum mechanics, which is itself the small-scale refinement of classical mechanics). The complete contents of my academic webpage remain available below. In particular, see here for more information on my graduate-level textbook Higher Algebra: Chapter 0, which is contracted to be published by Cambridge University Press in their Studies in Advanced Mathematics series.

I wrote a short memoir about my journey through academia entitled On Life and Math, which I posted on Substack starting here.

In my last math lecture I gave a short farewell speech, a transcript of which you can read here.

Please see here for info regarding exiting academia.

miscellanea

You can contact me at the Gmail address with username "the.etale.site".

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

This website is named after the mathematical concept of the étale site, a bold and beautiful reimagination of the geometric notion of "space". The étale site was introduced by the legendary mathematician Alexander Grothendieck as a means of applying the tools of geometry in the context of arithmetic (e.g. the study of prime numbers). The étale site comes in two flavors: the "small" one (which may be thought of as being "self-contained" internally within a given space), and the "large" one (which codifies the relationships between all of the different spaces that exist).

The Small Étale Site: Academic Mathematics (Past)

update 2025-03-20: I am making my 2020 NSF grant proposal freely available here. Although my collaborators and I have already essentially accomplished the main goal (Problem E) in paper #21 below, there are many beautiful ideas contained therein that remain open for further exploration. Most notably, I would be truly thrilled to see others establish a version of our braided (∞,2)-category using Schubert-stratified flag varieties organized over the Ran space of ℝ², as indicated in §2. See also this complementary document for a short summary of some further directions for exploration stemming directly from our paper (which aside from that of §3.4 are not present in the NSF grant proposal).

quick links

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textbook

• Higher Algebra: Chapter 0
  In progress (~95% complete -- and still to-be-completed!!). Contracted to be published by Cambridge University Press in their Studies in Advanced Mathematics series.

Broadly speaking, my research centered 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. I received a number of grants to support my research (over $300k in total), most notably a grant from the National Science Foundation entitled Factorization homology and low-dimensional topology (DMS-2105031).

[most info]       [least info]       ^♣ = student advisee

1. Symmetries of the cyclic nerve, with David Ayala and Nick Rozenblyum, 05/06/2024
   Advances in Mathematics, to appear.
   arxiv:2405.03897, 78 pages.

2. A braided (∞,2)-category of Soergel bimodules, with Yu Leon Liu^♣, David Reutter, Catharina Stroppel, and Paul Wedrich, 01/05/2024
   arxiv:2401.02956, 143 pages.
   [more info]
   Here is a video of a talk about this material (starting a minute or two in, due to a technical glitch). Updated slides are here -- those contain a (very sketchy) sketch of the proof, which I didn't discuss in the recorded talk.

   This paper is the primary research product of my NSF grant, the proposal for which I am making freely available; see above for that along with a complementary document summarizing some other further directions stemming directly from this paper.

   Here is a video of Catharina's 2022 ICM plenary address (whose accompanying survey article is here). She gives a broad overview of TQFT and categorification, and ends by putting our main theorem into that context.

   [less info]

3. Perverse schobers and 3d mirror symmetry, with Benjamin Gammage and Justin Hilburn, 02/14/2022
   Journal of the European Mathematical Society, to appear.
   arxiv:2202.06833, 43 pages.
   [more info]
   Here is a video of a talk by Justin about this material.

   [less info]

4. Derived Mackey functors and C_pⁿ-equivariant cohomology, with David Ayala and Nick Rozenblyum, 05/06/2021
   arxiv:2105.02456, 84 pages.
   [more info]
   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).

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

   [less info]

5. A universal characterization of noncommutative motives and secondary algebraic K-theory, with Reuben Stern^♣, 04/08/2021
   Annals of K-Theory, to appear.
   arxiv:2104.04021, 76 pages.
   [more info]
   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).

   [less info]

6. 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.

7. Stratified noncommutative geometry, with David Ayala and Nick Rozenblyum, 10/31/2019
   Memoirs of the AMS, 297 (2024), no 1485, iii+260 pp.
   arxiv:1910.14602, 236 pages.
   [more info]
   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 is a video of a talk by me that focuses on reflection (slides here), and here is one of one by David.

   [less info]

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

9. Goerss--Hopkins obstruction theory for ∞-categories, 12/18/2018
   Advances in Mathematics, to appear.
   arxiv:1812.07624, 54 pages.

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

    [less info]

11. Factorization homology of enriched ∞-categories, with David Ayala, John Francis, and Nick Rozenblyum, 10/17/2017
    arxiv:1710.06414, 45 pages.
    [more info]
    Here is a video of a talk about this material. Slides here.

    [less info]

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

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

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

15. 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.

16. 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.

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

18. 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.

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

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

21. 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.
    [more info]
    For the record, I would prefer an Oxford comma in the journal name, but I suppose it's not my prerogative to change it.

    [less info]

22. 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).
   [more info]
   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.

   [less info]

-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.)

miscellanea

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. 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, Dohan, Duhl-Coughlin, Farris, Gepner, Hansen, Jordan, Li-Bland, Nadler, Orman, Penneys, Reutter, Roberts, Spivak, Treumann, White, Yetter.