Research

We compute the cooperation algebras for motivic integral cohomology, algebraic K-theory, and hermitian K-theory over all p-adic fields and the rationals. We also construct spectrum-level splittings for the motivic integral cohomology and algebraic K-theory cooperations, and provide a partial decomposition of the hermitian K-theory cooperations.

We construct a $w_1$-self-map of periodicity 1 on the $\mathbb{C}$-motivic analogue of double of Davis and Mahowald's spectrum $\mathrm{Y}$, denoted $\mathrm{DY}$. The cofiber of this self-map is a realization of $\mathrm{DA}(1)$, the double of the subalgebra $A(1)$ of the $\mathbb{C}$-motivic Steenrod algebra.

In this survey, we study how tools from stable homotopy theory have manifested and impacted motivic homotopy theory. In particular, we discuss various motivic Adams spectral sequences, periodicity in the motivic stable homotopy groups of spheres, and synthetic spectra. We conclude with many problems for future investigation.

We compute the Atiyah Real K-theory of $C_2$-equivariant projective spaces and construct immersions of such spaces into multiples of the regular representation. These computations are made tractable by the recent geometric filtration of equivariant projective spaces due to Bhattacharya-Waugh-Zeng-Zou, together with a variant of the localized slice spectral sequence introduced by Meier-Shi-Zeng. As an immediate corollary of these computations, we obtain an equivariant analogue of James periodicity.

We construct spectrum-level splittings of $\mathrm{BPGL}\langle 1 \rangle \wedge \mathrm{BPGL}\langle 1 \rangle$ at all primes $p$, where $\mathrm{BPGL}\langle 1 \rangle$ is the first truncated motivic Brown--Peterson spectrum. Classically, $\mathrm{BP}\langle 1 \rangle \wedge \mathrm{BP}\langle 1 \rangle$ 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 $\mathrm{char}(\mathbb{F}_q) \neq p$. As an application, we compute the $\mathrm{E}_1$-page of the $\mathrm{BPGL}\langle 1 \rangle$-based Adams spectral sequence as a module over $\mathrm{BPGL}\langle 1 \rangle$, both in homotopy and in terms of motivic spectra. We also record analogous splittings for $\mathrm{BPGL}\langle 0 \rangle \wedge \mathrm{BPGL}\langle 0 \rangle$.

We compute the ring of cooperations for the very effective hermitian K-theory over all finite fields $\mathbb{F}_q$ where $\mathrm{char}(\mathbb{F}_q)\neq 2$. 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 $\mathrm{E}_1$-page of the $\mathrm{kq}$-resolution.

Let $\mathrm{kq}$ denote the very effective cover of the motivic hermitian K-theory spectrum. We analyze the ring of cooperations for $\mathrm{kq}$ in the stable motivic homotopy category $\mathrm{SH}(\mathbb{R})$, giving a full description in terms of Brown–Gitler comodules. To do this, we decompose the $\mathrm{E}_2$-page of the motivic Adams spectral sequence and show that it must collapse. The description of the $\mathrm{E}_2$-page is accomplished by a series of algebraic Atiyah–Hirzebruch spectral sequences which converge to the summands of the $\mathrm{E}_2$-page. Along the way, we prove a splitting result for the very effective symplectic K-theory $\mathrm{ksp}$ over any base field of characteristic not two.

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.