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; 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.
Recently I have been finishing up the foundations of model ∞-categories, which provide a robust theory of resolutions that is native to the ∞-categorical setting. These foundations are contained in the following papers.

main 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
supplementary papers
  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
For background on the original motivation for this project -- namely, equivariant & motivic Goerss--Hopkins obstruction theory --, see here. However, I am also hopeful that this framework will be useful elsewhere. For example, the Kan--Quillen model structure on the ∞-category of simplicial spaces accommodates a host of methods for proving that limits of simplicial spaces commute with geometric realization, which should be useful in algebraic K-theory (via Waldhausen's S-dot construction). Moreover, the resolution model structure (a/k/a the "E2 model structure") presents the nonabelian derived ∞-category, which plays a prominent role for instance in Barwick's universal characterization of algebraic K-theory as well as in his theory of spectral Mackey functors (which present the ∞-category of genuine equivariant spectra).
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,...].