This webpage is a place for me to organize my papers into projects and to provide a bit of explanation for them, to clarify how my individual publications fit into continuing broader projects with overarching goals. Most of these papers are also available on the preprint arXiv and, for the ones that have been published already, on the journal websites as well. I haven't marked the publication status of each paper on this page, because that isn't what this page is for. If you want to know whether a given paper is published, and in what journal, a web search will tell you that. Alternatively, you can consult ORCID, which is usually pretty up-to-date on published papers.

Versions on this page may be more current (with corrections of minor typos which don't warrant a full arXiv update) than the arXiv versions.
  1. "Complex multiplication and homotopy groups of spheres" project, with subtopics:

  2. Other papers in pure mathematics

  3. Papers on data analysis and fMRI

  4. Notes

"Complex multiplication and homotopy groups of spheres" project.

This is the main project I work on. The basic theme is to describe the orders of the stable homotopy groups of Bousfield-localized finite spectra in terms of special values of L-functions and zeta-functions that arise in number theory. The broad motivation for such a project is that it is generally extremely difficult to even describe the v_n-periodic families in the homotopy groups of spheres for n>1, so when you do difficult computations in this area, even the statements of the theorems (let alone their proofs!) are almost always frustratingly long and difficult to parse. Sometimes you can get a simpler and more satisfying statement by describing the orders of the relevant homotopy groups in terms of something else that's also difficult to understand but at least well-studied elsewhere in mathematics--in particular, special values of some L-function or zeta-function.

In this project I have often found it useful it to use formal groups with complex multiplication, that is, "formal modules." For a number field K, some of the flat cohomology groups of the moduli stack of formal O_K-modules have orders describable in terms of the Dedekind zeta-function of K (e.g. see the "Structure and cohomology..." paper, below). Formal modules also expedite some "height-shifting" tricks, since if K is a degree d extension of the p-adic rationals, then the automorphism group of an O_K-height n/d formal O_K-module naturally embeds into the height n full Morava stabilizer group as a rather large profinite subgroup. This is exploited in various ways in the papers below, particularly in "Moduli of formal A-modules under change of A" and "Height four formal groups..." and "The cohomology of the height four...".

The project is far from finished, but so far I have been able to make progress in several directions, which you can read about in the papers below. I have also been (slowly) working on a book on this material which is intended as a companion piece to Ravenel's book "Complex cobordism and stable homotopy groups of spheres."

The papers in this project fit into several threads, which interact with each other:

Other papers in pure mathematics.

Papers on data and fMRI.

For several years I have worked on applications of methods from higher mathematics to fMRI data, in collaboration with the Diwadkar lab in WSU's medical school, as well as with my students and collaborators within the math department. This is a long way from the pure mathematics projects I largely focus on, but I suspect it's probably good for a researcher to think about a mix of pure questions and applied questions.

In the beginning our group's focus was narrowly on the application of topological tools (particularly persistent homology) to fMRI, but more recently we have taken a broader view, using simplicial methods from topology as well as techniques of statistical inference to develop some new tools for higher-order network discovery and analysis. The papers we have finished so far are below.

Notes.

Luca Candelori and I organized a weekly seminar on modular forms, here in the mathematics department at WSU, during the 2018-2019 academic year. One major goal of the seminar was to cover enough background from number theory and topology for us to discuss two open, probably rather difficult, questions: 1. whether one can give K-theoretic or homotopy-theoretic descriptions of special values of L-functions of eigenvalue 1/4 Maass cusp forms, and 2. whether there exist spectral enrichments of geometric models for Maass forms (of some fixed level and eigenvalue), i.e., whether there exist "topological Maass forms," especially in the harmonic case and the eigenvalue 1/4 case.

Below are some of our notes from the seminar. We made some effort to include the number-theoretic content that readers with a background in homotopy theory are less likely to already know, and the homotopy-theoretic content that readers with a background in number theory are less likely to already know.