Research/Writings

My research interests are in homotopy theory, motivated from the chromatic perspective, and the tools I use are often computational. I have recently been interested in periodicity in motivic and equivariant stable homotopy theory, and I like to use the Adams spectral sequence to access periodic elements (although recently, I have had an eye towards the slice spectral sequence…).

Preprints/Publications

• A $\mathbb{C}$-motivic $w_1$-self-map of periodicity 1, M. In preparation.
  Abstract

  We construct a finite $\mathbb{C}$-motivic spectrum with a $w_1$-self-map of periodicity 1. The cohomology of the cofiber of this self-map has a module structure over the Steenrod algebra which we use to create splittings of many chromaticcaly interesting motivic spectra.
• Splittings of truncated motivic Brown-Peterson cooperations algebras, M., Sarah Petersen, and Liz Tatum. Submitted. arxiv:2509.19542
  Abstract

  We construct spectrum-level splittings of $\text{BPGL}\langle 1\rangle\wedge \text{BPGL}\langle 1\rangle$ at all primes $p$, where $\text{BPGL}\langle 1 \rangle$ is the first truncated motivic Brown-Peterson spectrum. Classically, $\text{BP}\langle 1 \rangle \wedge \text{BP}\langle 1\rangle$ was first described by Kane and Mahowald in terms of Brown-Gitler spectra. This splitting was subsequently reinterpreted by Lellman and Davis-Gitler-Mahowald in terms of Adams covers. In this paper, we give motivic lifts of these splittings in terms of Adams covers, over the base fields $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{F}_q$, where $\text{char}(\mathbb{F}_q)\neq p$. As an application, we compute the $E_1$-page of the $\text{BPGL}\langle 1 \rangle$-based Adams spectral sequence as a module over $\text{BPGL}\langle 1 \rangle$, both in homotopy and in terms of motivic spectra. We also record analogous splittings for $\text{BPGL}\langle 0 \rangle \wedge \text{BPGL} \langle 0 \rangle$.
• Rings of cooperations for Hermitian K-theory over finite fields, M. Submitted. arxiv:2509.02786
  Abstract

  We compute the ring of cooperations $\pi_{**}(\text{kq} \otimes \text{kq})$ for the very effective Hermitian K-theory over all finite fields $\mathbb{F}_q$ where $\text{char}(\text{F}_q) \neq 2$. To do this, we use the motivic Adams spectral sequence and show that all differentials are determined by the integral motivic cohomology of $\mathbb{F}_q$. As an application, we compute the $E_1$-page of the kq-resolution.
• On the ring of cooperations for real Hermitian K-theory, M. Submitted. arxiv:2506.16672
  Abstract

  Let $\text{kq}$ denote the very effective cover of the motivic Hermitian K-theory spectrum. We analyze the ring of cooperations $\pi_{**}(\text{kq} \otimes \text{kq})$ in the stable motivic homotopy category $\text{SH}(\mathbb{R})$, giving a full description in terms of Brown-Gitler comodules. To do this, we decompose the $E_2$-page of the motivic Adams spectral sequence and show that it must collapse. The description of the $E_2$-page is accomplished by a series of algebraic Atiyah-Hirzebruch spectral sequences which converge to the summands of the $E_2$-page. Along the way, we prove a splitting result for the very effective symplectic K-theory $\text{ksp}$ over any base field of characteristic not two.
• Toric double determinantal varieties, Alexander Blose, Patricia Klein, Owen McGrath, M. Published in Communications in Algebra 49 (2021), 7, 3085-3093. arxiv:2006.04191
  Abstract

  We examine Li's double determinantal varieties in the special case that they are toric. We recover from the general double determinantal varieties case, via a more elementary argument, that they are irreducible and show that toric double determinantal varieties are smooth. We use this framework to give a straighforward formula for their dimension. Finally, we use the smallest nontrivial toric double determinantal variety to provide some empirical evidence concerning an open problem in local algebra.

Writings

• Splittings and the algebraic Atiyah-Hirzebruch spectral sequence Expanded set of notes for preseminar talk at UW. PDF
• Galois descent and the Picard group of K-theory Notes for a talk in the UW Student AG Seminar. PDF
• The Adams spectral sequence and Hopf algebroids Notes for a talk in the DUBTOP seminar. PDF

The ring of cooperations $\pi_{**}(\text{kq} \otimes \text{kq})$ over $\mathbb{F}_3$.

Recent Talks

• Cooperations in motivic homotopy theory, JMM, January 2026. Slides
• Splittings and cooperations in motivic homotopy theory, University of Virginia Topology Seminar, November 2025
• Splittings and cooperations in motivic homotopy theory, University of Kentucky Topology Seminar, November 2025
• Higher Adams differentials and hidden extensions, eCHT Secondary Steenrod Algebra Reading Seminar, November 2025
• Splittings and cooperations in motivic homotopy theory, Duke University Geometry and Topology Seminar, October 2025
• Splittings of truncated motivic Brown-Peterson cooperations algebras, University of Colorado Boulder Homotopy Theory Seminar, September 2025
• On the ring of cooperations for real Hermitian K-theory, University of Michigan Geometry Seminar, September 2025
• Splittings and cooperations in motivic homotopy theory, University of Notre Dame Topology Seminar, September 2025
• On the ring of cooperations for real Hermitian K-theory, University of Washington Topology Seminar, June 2025

More detailed research interests.

My research is oriented by chromatic homotopy theory. Roughly speaking, here is the story. As a homotopy theorist, the one thing I really want to understand more than anything else are the homotopy groups of spheres. There are some reductions/refinements one can make:

• Freudenthal tells us that these groups stabilize, so let’s focus on computing the stable homotopy groups of spheres;
• Serre tells us that the non-negative stable stems are finite abelian groups, so let’s work one prime at a time;
• Morava (or maybe God) tells us that at a fixed prime, the stable stems are organized into $v_n$-periodic families, so let’s focus on computing one periodic family at a time.

There is a lot of fun one can have now in $v_n$-periodic homotopy theory. For example, there are two localization functors which allow one to access these periodic families, one more algebraic and one more topological, and the telescope conjecture equates these two localizations. Famously, the telescope is false at all primes and all heights 2 and above. However, at the height 1, and say now at the prime 2, Mahowald (or maybe God) used like a zillion spectral sequences and computed the $v_1$-periodic stable stems entirely and showed that the two aforementioned localizations actually agree at height 1, proving the height 1 telescope conjecture. This is the motivation behind most of my current research: to try to understand periodicity in other contexts by means of a zillion spectral sequences, and more generally to understand the telescope conjecture and other cool chromatic things in other contexts.

Most of my research has been housed in motivic homotopy theory, a variant of stable homotopy theory which incorporates algebraic-geometry over a fixed base scheme. One can ask for a hands on approach to $v_1$-periodicity in motivic homotopy theory, and my current research studies this at all primes over the complex numbers, the real numbers, and finite fields. However, the motivic stable stems are much more interesting than the classical stable stems. I am interested in studying the following:

• determining exotic periodic elements in the motivic stable stems and, more generally, better understanding exotic periodicity;
• understanding the correct motivic analogue of the telescope conjecture, one which captures classical and exotic periodicity;
• moving towards a more integral class of base scheme.

Most of my current work has used the Adams spectral sequence as a guiding principle (although recently, I have had an eye towards the slice spectral sequence…). I am particularly fond of comparing and contrasting these results as the base scheme varies and with classical stable homotopy theory, and I am eager to connect my work over the real numbers with $C_2$-equivariant homotopy theory.