Research

We construct a plethora of finite motivic spectra and identify their minimal periodic self-maps.

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.

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 C2-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 BPGL⟨1⟩∧BPGL⟨1⟩ at all primes p, where BPGL⟨1⟩ is the first truncated motivic Brown--Peterson spectrum. Classically, BP⟨1⟩∧BP⟨1⟩ 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 ℂ,ℝ, and 𝔽q, where 𝔽q≠p. As an application, we compute the E1-page of the BPGL⟨1⟩-based Adams spectral sequence as a module over BPGL⟨1⟩, both in homotopy and in terms of motivic spectra. We also record analogous splittings for BPGL⟨0⟩∧BPGL⟨0⟩.

We compute the ring of cooperations for the very effective hermitian K-theory over all finite fields 𝔽q where char(𝔽q)≠2. To do this, we use the motivic Adams spectral sequence and show that all differentials are determined by the integral motivic cohomology of 𝔽q. As an application, we compute the E1-page of the kq-resolution.

Let kq denote the very effective cover of the motivic hermitian K-theory spectrum. We analyze the ring of cooperations for kq in the stable motivic homotopy category SH(ℝ), giving a full description in terms of Brown–Gitler comodules. To do this, we decompose the E2-page of the motivic Adams spectral sequence and show that it must collapse. The description of the E2-page is accomplished by a series of algebraic Atiyah–Hirzebruch spectral sequences which converge to the summands of the E2-page. Along the way, we prove a splitting result for the very effective symplectic K-theory 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.

Expanded set of notes for DUBTOP seminar talk.</em>

Expanded set of notes for preseminar talk at UW.
Notice: Section 7.1 has an incorrect E(1)-comodule structure on the Brown-Gitler comodule.

Notes for a talk in the UW Student AG Seminar.
Notice: Example 3.2 and Example 4.1 are incorrect! It is not true that Galois extensions of number fields induce Galois extensions of their rings of integers. This is precisely why Example 4.1 is so hard; I forgot about ramification.

Notes for a talk in the DUBTOP seminar.