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.
- Model ∞-categories I: some pleasant properties of the ∞-category of simplicial spaces (substantial updates to v2)
- The universality of the Rezk nerve
- All about the Grothendieck construction
- Hammocks and fractions in relative ∞-categories
- Model ∞-categories II: Quillen adjunctions
- Model ∞-categories III: the fundamental theorem
In turn, these are supported by the following supplementary papers (which do not appear in my thesis).
- A user's guide to co/cartesian fibrations
- From fractions to complete Segal spaces, joint with Zhen Lin Low
- 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[{x_{n}:n≠2^{t}-1}]=Z/2[x_{2},x_{4},x_{5},x_{6},x_{8},x_{9},...].
The complex cobordism ring is π_{*}(MU)=Z[{x_{2n}}]=Z[x_{2},x_{4},x_{6},...].
a/s/l?