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
- 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
supplementary papers
- 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
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 "E^{2} 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[{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?