Aaron Mazel-Gee

My name is Aaron. I like to do math. It is my job. I'm a grad student at UC Berkeley. I study algebraic topology. My advisor is Peter Teichner.

My email address is edu.berkeley.math@aaron, except that that's not quite it. My office at Berkeley is 1045 Evans.

Throughout 2012, I was based at the Max Planck Institute for Mathematics in Bonn, Germany. Throughout 2013, I was based at MIT in Cambridge, MA. During the fall 2015 semester, I was based at Montana State University in Bozeman, MT, where I worked with David Ayala.

You can see my c.v. here.

My current research interests are: factorization homology, derived algebraic geometry, and algebraic K-theory; quantum field theory and shifted geometric 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.
My PhD thesis is entitled Goerss--Hopkins obstruction theory via model ∞-categories. You can see it here.* In it, I 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. Using this, I generalize Goerss--Hopkins obstruction theory to an arbitrary (presentably symmetric monoidal stable) ∞-category. As a sample application, I use this generalized obstruction theory to construct E structures on the motivic Morava E-theory spectra; 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).

*Last updated 5/19/2016. I am happy to receive any comments, errata, typos, etc.

I have also 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.

The first part of my thesis (regarding model ∞-categories) is assembled from the following papers.

  1. Model ∞-categories I: some pleasant properties of the ∞-category of simplicial spaces (substantial updates to v2)
  2. The universality of the Rezk nerve
  3. All about the Grothendieck construction
  4. Hammocks and fractions in relative ∞-categories
  5. Model ∞-categories II: Quillen adjunctions
  6. Model ∞-categories III: the fundamental theorem
In turn, these are supported by the following supplementary papers (which do not appear in my thesis).
  1. A user's guide to co/cartesian fibrations
  2. From fractions to complete Segal spaces, joint with Zhen Lin Low
  3. Quillen adjunctions induce adjunctions of quasicategories

The *actual* final version of 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.
classes       xkcd seminar       teaching       talks       conferences       writing       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,...].