Aaron Mazel-Gee

My name is Aaron. I like to do math. It is my job. I recently finished grad school at UC Berkeley. I am currently a Research Assistant Professor at USC.

You can see my c.v. here.

My email address is etale.site@aaron, except that that's not quite it.

My office at USC is KAP 438D.

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

My current research interests are: factorization homology, derived algebraic geometry, and algebraic K-theory; Fukaya categories, arboreal singularities, and microlocal sheaves; duality phenomena in quantum field theory via factorization homology; quantum field theory and shifted symplectic/Poisson structures; higher category theory, abstract homotopy theory, and their applications to equivariant and motivic homotopy theory; chromatic homotopy theory and its interactions with number theory; the human condition.

The geometry of the cyclotomic trace: In collaboration with David Ayala and Nick Rozenblyum, I have recently developed a new construction of topological cyclic homology (TC), which 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:

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.)

You can also see a video of a talk about this material here.
Slides (updated from the ones appearing in the video) are available here (and in "handout" format (so with far fewer slides) here).
Slides containing much more detail (from 3 hours' worth of talks) are available here (and in "handout" format 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).

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 2/19/2018. 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 12 referenced above (click "show more" to see those related to my thesis)).

A relative Lubin--Tate 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. [abstract]

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 K-theoretic localizations of Morava E-theory corepresent a version of the Lubin--Tate moduli problem in this framework.

[hide abstract]
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. [abstract]

abstract: The classical Lusternik-Schnirelman-Borsuk theorem states that if a d-sphere 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 d-cube 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 d-cube by d sets of ridges, at least one set must contain a pair of antipodal k-faces, and we determine the maximum k for which this must occur, for all dimensions except d=5.

[hide abstract]
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, Ben-Zvi, Carchedi, Corwin, Duhl-Coughlin, Farris, Gepner, Hansen, Jordan, Li-Bland, Nadler, Orman, Penneys, Roberts, Spivak, Treumann, White.

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[x₂,x₄,x₅,x₆,x₈,x₉,...].
The complex cobordism ring is π_{_*}(MU)=Z[{x_2n}]=Z[x₂,x₄,x₆,...].
a/s/l?