Department of Mathematics

Research Interests

I am interested in the more algebraic aspects of algebraic topology. In particular, I spend some time thinking about:

• Homotopy groups of spheres and how they are organized by the chromatic filtration.
• The relationship between the chromatic localization of a spectrum and its 0th space.
• Rigidifying multiplicative structures on ring spectra to H-Infinity and E-Infinity structures.

Dissertation

In addition to the components listed in the section "Works in Progress" my thesis contains the following.

I have studied the complex-oriented cohomology ring of BU (with the additional structure coming from the direct sum and tensor product operations on vector bundles) by looking at the associated formal scheme.This formal scheme can be identified with a scheme of a version of the Witt vectors, with a multiplicative structure related to the formal group law associated to our generalized cohomology.

Using this identification, the Husemoller-Witt splitting of the cohomology of BU, the splitting of the Big-Witt scheme and Quillen's splitting of MU are all realized simultaneously (i.e. one construction produces all of them) over a p-local ring.

In this setting, we can compute the action of the Cartier ring on our Hopf-algebra. In particular, we can compute the action of the Verschiebung, Frobenius and homothety operators. Along the way we write down formulas for the generalized Chern classes of a tensor product of vector bundles.

My thesis is still in progress and is available by

Slides

I have given several talks on this topic of varying length and detail:

I gave a full length talk at the Algebraic Topology Conference in Buenos Aires, Argentina. Here are the slides.

I gave shorter talks on this at the AMS Regional Conferences in Kalamazoo, MI and Vancouver, BC. Here are those slides.

I also gave an introductory talk focused on the tensor product structure at the 2008 Graduate Student Topology Conference at UIUC. Here are those slides.

Work in Progress

H-Infinity Ring Structures on BP

Joint with Niles Johnson

Draft in Progress: H-Infinity Structures on BP and the frontend to the the accompanying Mathematica module.

Quillen's idempotent operation on p-local MU splits this spectrum into a wedge of suspensions of the Brown-Peterson spectrum. The geometric structure defining MU gives it the structure of an E-Infinity ring spectrum. This gives p-local MU an E-Infinity ring structure and it is natural to ask if BP can be given a compatible E-Infinity ring structure.

The Brown-Peterson spectrum plays a special role in the calculation of homotopy groups of spheres and an E-Infinity ring structure would give a number of convenient tools for computations. For these purposes, the weaker notion of an H-Infinity ring structure would suffice.

I have given a necessary and sufficient condition for BP to have an H-Infinity ring structure compatible with the projection map r, from p-local MU to BP using the language of formal group laws. Namely, BP has a compatible H-Infinity-ring structure if and only if the formal group law F, defined below, is a p-typical formal group law.

To construct F, we first show that the Z/p-Borel-equivariant form of BP is a Landweber-exact multiplicative cohomology theory on trivial Z/p spaces. Then we show that there is a degree 2p formal group law F on this theory defined by the applying the cyclic-power operation to p-local MU and then applying Quillen's map r.

McClure has given a purely computational criterion for BP to be a H-Infinity as well. Niles Johnson and I have simplified his formulas dramatically. McClure's condition defines a map theories from MU to Z/p-Borel-equivariant Brown-Peterson cohomology and an associated map of formal groups. Through this we see his condition is equivalent to ours.

Our computer calculations have shown, when p=2 or 3, Quillen's map is NOT H-Infinity and hence not E-Infinity. We also have explicit calculations of these power operations on MU_* through a range.

In a related vein, we have shown that E_1 does not admit a p-typical H-Infinity orientation for any prime p. This implies that, 2-locally, KU and its connective cover ku do not admit p-typical H-Infinity orientations.

The Morava K-theory of Coker J and Higher Chromatic Analogues

Joint with Nick Kuhn

We show that the K(2)-homology of the infinite loop space Coker J is isomorphic to the free Morava K-theory Dyer-Lashof algebra on a single generator. Corollaries of this computation include a description of the E-cohomology of Coker J, where E is either K(2), E_2, tmf, or E_2^{hG}, where G is a subgroup of the second Morava stabilizer group. Then we define higher chromatic analogues of Coker J to derive analogous results. This extends the work of Hodgkin and Snaith on the K-theory of the infinite loop spaces J and Coker J.

Computing the Homology of a DGA using Mathematica

I have written a Mathematica program for computing the May spectral sequence (as an algebra and in characteristic 2) through a user-defined range. This takes in as input the differentials (which are not hard to compute for this example) and does the chugging of computing the homology groups, finding a basis and determining all the generators and relations in the given range.

The method applied is general and applies to any spectral sequence of algebras over a field that is finite dimensional in each bi-degree. In fact it can be made significantly faster for characteristic 0 fields.

The primary purpose of this program is to ensure that we have all generators and relations as an algebra. It can assist in hand computations when things get hairy enough that errors keep cropping up.