Research and Preprints

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My research focuses on computational aspects of stable homotopy theory. My thesis covered methods for computing with the conjectural "rings of integers" in the Hopkins-Miller spectra EO_{p-1}. My recent research has focused on building these appropriate connective spectra, finding and understanding the homotopy of EO_{p^k(p-1)} for k bigger than 0, understanding twisted tmf homology at various primes, and computing THH of small, well behaved spectra.

Many of these works are very much works in progress. I ask that you forgive misspellings, misstatements, mistakes, and omissions. If a particular topic is missing its corresponding document, then it has not been posted because either it is a work in progress with collaborators or I am working to bring it into a readable form.

The dates in parentheses are the date of most recent update for works in progress.

Published and Accepted

The tmf Homology of BSigma3 In this paper, I give a simple Hopf algebra usable for computing the tmf homology of a space or spectrum at the prime 3. To demonstrate the effectiveness of this, I mirror work of Mahowald and Milgram and compute the tmf homology of the classifying space of the symmetric group on 3 letters. Rounding out computations, I give the tmf homology of the finite skeleta of the cofiber of the transfer BSigma_3->S^0. This appeared in Proc AMS

Cyclic comodules, the Homology of j and j-Homology This short paper arose as a warm up to a project with Mark Behrens. Using general techniques for comodules and coalgebras over a Hopf algebra, I give elementary proofs of the homology of the connective j spectrum at p=2 and p odd. I also compute the j-module dual Steenrod algebra, the homotopy of HF_p smashed with itself over j. This appeared in Topology and Its Applications

The 5-Local Homotopy of eo_4 Using a Hopf algebroid developed by Hopkins, I run a series of Bockstein spectral sequences to compute the 5-local homotopy of the conjectural spectrum eo_4. The computation allows for a complete description of the analogue of the ring of modular forms at this height, and it allows other conjectures to be checked readily. Moreover, if we invert a distinguished element, we recover the homotopy of eo_{p-1}[Delta^{-1}]. This is to appear in Algebraic and Geometric Topology

A v_2^32 self-map of M(1,4) at the prime 2 (Joint with M. Behrens, M. Hopkins, and M. Mahowald). Hopkins and Mahowald showed that the generalized Smith-Toda complex M(1,4) at the prime 2 admits a v_2^32 self map. Behrens and I are fleshing out their arguments, providing all of the details and giving an overview of Ext computations with Brown-Gitler modules. This is to appear in Homology, Homotopy, and Applications.

The String Bordism of BE_8 The contains a computation of the low dimensional String bordism groups. I use the Adams spectral sequence for the primes 2 and 3 to find the structure of the group, comparing String bordism with tmf via the sigma orientation.

Submitted

THH of \ell and ko (Joint with V. Angeltveit and T. Lawson) Vigleik, Tyler and I work out the integral homotopy groups of the topological Hochschild homology of ku. This supplements the known results about the homotopy of THH(ku;HF_p) and of THH(ku;ku/p). We also discuss conjectural results about what happens when one considers BP<n> instead of ku.

The spectra ko and ku are not Thom spectra (Joint with V. Angeltveit and T. Lawson). Using our computation of THH(ko) and THH(ku) together with recent work of Blumberg et al, we derive a simple argument that neither ku nor ko is a Thom spectrum.

Preprints

The action of the Morava Stabilizer Group on the Lubin-Tate spectrum (Joint with M. Hopkins and D. Ravenel). We have a new argument to prove that if G is a finite subgroup of the Morava stabilizer group, then the action of G on the Lubin-Tate ring of deformations has a particularly nice description. In particular, the homotopy fixed point spectral sequence for the Hopkins-Miller higher real K-theories becomes more manageable. This project falls naturally into three pieces: the action of the finite subgroups, the computation of the differentials, and applications.

Motivic Computations

These are currently being completely rewritten!

Homological obstructions to string orientations (Joint with C. Douglas and A. Henriques). We show that manifolds with certain orientations posess a stronger form of Poincare duality reflected in the Steenrod operations. This provides a primary obstruction to higher orientations that is more readily checked than computing characteristic classes.

A Hopf algebra for Computing eo_{p-1} Homology In this work in progress, I describe the ways the results about ko at the prime 2 and tmf at the prime 3 generalize to arbitrary primes. Along the way, I introduce spectra eu_{p-1} which are to eo_{p-1} what ku is to ko. I also find a Hopf algebra which allows computing with these spectra, and I use it to compute the eo_{p-1} homology of BSigma_p. I also prove a splitting result similar to Mahowald and Milgram's of ku smashed with the cofiber of the transfer BSigma_2->S^0.

The Homology of the spectrum q(2) (Joint with M. Behrens). This program is part of an approach to the telescope conjecture at height 2, using information gleaned from the two halves of the K(2)-local sphere described by Behrens. We currently work to understand the two most important cases: p=2 and p=3.

My Thesis This contains many of the results which appear in the above preprints. It contains additionally a result linking higher differentials in the spectral sequence of a filtered algebra to Massey products on lower differentials, and I also compute the Tate cohomology of Z/p with coefficients in an associated graded of homotopy of E_{p^k(p-1)}. This serves as a starting point for computing the homotopy fixed points.

Odds and Ends
Slides and Write-ups from Talks Pie in the Sky Projects
  • The twisted tmf homology of the classifying spaces of simple Lie groups (Joint with Chris Douglas).
  • Construction of the spectra eo_{p-1} (Joint with Jacob Lurie).
  • Generalizations to all heights of the Hopf algebras for eo_{p-1}.
  • Generalizing bo resolutions at 2 to all primes using eo_{p-1}.
  • A Massey Product Primer.
Useful Computations