My email address is edu.berkeley.math@aaron, except that that's not quite it. My office at Berkeley is 741 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.

My current research interests are: higher category theory, abstract homotopy theory, and their applications to equivariant and motivic homotopy theory; chromatic homotopy theory and its interactions with number theory; derived algebraic geometry; the human condition.

My current projects include...

- ...a hunt for manifestations of certain properties of
*p*-adic modular forms in*tmf*. The idea is that*p*-adic modular forms, despite being inherently*K(1)*-local, should nevertheless be able to detect the existence of the*K(2)*-local world. This might be described as an investigation of "the class field theory of*tmf*". You can find a very rough sketch here, and here is a fun (although by now somewhat outdated) little flowchart that I've been keeping to help me organize how all the various concepts relate to each other. You can also find a somewhat more developed perspective on this -- in particular, an explanation of the connection to class field theory -- here (towards the end). - ...an ∞-categorical Goerss--Hopkins obstruction theory. Given a symmetric monoidal ∞-category
*C*equipped with some notion of*E*-homology and given an ∞-operad*O*such that*E*-homology takes*O*-algebras in*C*to algebras for some monad Φ, this can be used to determine (the homotopy groups -- and in particular the emptiness or nonemptiness -- of) the moduli space of those objects of*C*with*E*-homology isomorphic to any given Φ-algebra*A*. Moreover, these computations are entirely algebraic: the obstruction groups are given by André--Quillen cohomology in the category of Φ-algebras. - ...a theory of "genuine ∞-operads" suitable for encoding "genuine commutativity" for equivariant and motivic ring spectra. For instance, the category of
*G*-spectra is a*G*-monoidal category, and so whereas an ordinary commutative*G*-equivariant ring spectrum is a*G*-spectrum*R*equipped with coherent multiplication maps*R*, a genuine commutative^{∧n}→R*G*-equivariant ring spectrum ought to have coherent multiplication maps*R*for all finite^{∧T}→R*G*-sets*T*. (There is a free functor from the former to the latter, but of course this does not describe all genuine commutative*G*-equivariant ring spectra.) This admits an analog in the motivic world, where for instance in the motivic stable category over a field*k*one can ask for multiplication maps*R*^{∧(Spec A)}→*R*for all étale*k*-algebras*A*. Moreover, étale realization should induce a functor from genuine commutative motivic ring spectra over*k*to genuine commutative Gal(*k**/k*)-equivariant ring spectra. This framework should also afford a clean ∞-categorical description of the global equivariant homotopy category (and suggest the appropriate motivic analog). - ...the pushout of the above two projects. Hopefully this will eventually include concrete calculations. For instance, there are a number of motivic spectra which one might hope carry essentially unique genuine commutative structures (the sphere spectrum, algebraic
*K*-theory, algebraic cobordism, Eilenberg--Mac Lane spectra, etc.). Extremely optimistically, this might even lead to a construction of "motivic modular forms" (that is, a motivic version of*tmf*). Of course, it's entirely possible that the relevant obstructions will already vanish even if we only work with motivic*E*_{∞}-ring spectra, but this will then ultimately only endow "*mmf*" with an*E*_{∞}-structure. - ...a categorical framework in which to quantify the extent to which some category captures "all the structure" on the image of some topology-to-algebra functor, especially for algebras over a monad. For instance, the
*p*-adic*K*-theory of an*E*_{∞}-ring spectrum is naturally a*θ*-algebra. Is this the best possible characterization? If so, why?

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.

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

a/s/l?