Most of these papers are also available on the preprint arXiv and journal websites as well. 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

  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 study formal groups with complex multiplication -- that is, "formal modules" -- and exploit some of their properties to make deeper (that is, higher height) computations in the stable homotopy groups of spheres, and to establish descriptions of the zeta-functions associated to a number field K in terms of the flat cohomology groups of the moduli stack of formal groups with complex multiplication by the ring of integers of K. The project is far from finished, but already there are advances made in both of these two directions, which you can read in the papers below. I have also been 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."

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.


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.