We compute the homotopy groups of the $\eta$-periodic motivic sphere spectrum over a finite-dimensional field $k$ with characteristic not $2$ and in which $-1$ is a sum of four squares. We also study the general characteristic 0 case and show that the $\eta$-periodic slice spectral sequence over $\mathbb{Q}$ determines the $\eta$-periodic slice spectral sequence over all extensions of $\mathbb{Q}$.

Joint with Oliver Röndigs

Submitted, 14pp.

For a finite group $G$, we introduce the complete suboperad $Q_G$ of the categorical $G$-Barratt-Eccles operad $P_G$. We prove that $P_G$ is not finitely generated, but $Q_G$ is finitely generated and is a genuine $E_\infty$ $G$-operad (i.e., it is $N_\infty$ and includes all norms). For $G$ cyclic of order $2$ or $3$, we determine presentations of the object operad of $Q_G$ and conclude with a discussion of algebras over $Q_G$, which we call biased permutative equivariant categories.

Joint with Kayleigh Bangs, Skye Binegar, Young Kim, Angélica M. Osorno, David Tamas-Parris, and Livia Xu

Submitted, 20pp.

We determine systematic regions in which the bigraded homotopy sheaves of the motivic sphere spectrum vanish.

Joint with Oliver Röndigs and Paul Arne Østvær

Forum Math. Sigma 6 (2018), e3, 20pp.

For a real closed field k, we prove that stable C₂-equivariant homotopy theory embeds fully faithfully into the stable motivic homotopy category over k. This "uncompletes" an $\eta$-complete version of this theorem from previous work.

Joint with Jeremiah Heller

New directions in homotopy theory, Contemp. Math. 707 (2018), 1-9

We study the tensor triangular geometry of the stable motivic homotopy category, proving that the Balmer map to the homogeneous Zariski spectrum of Milnor-Witt $K$-theory is surjective.

Joint with Jeremiah Heller

Geom. Topol. 22 (2018), no. 4, 2187-2218.

We relate three different schemes for understanding tmf∧tmf at the prime 2: bo-Brown-Gitler modules, 2-variable modular forms, and modular forms with level structure.

Joint with Mark Behrens, Nat Stapleton, and Vesna Stojanoska

Accepted for publication in J. Topology

We study some functors linking G-equivariant homotopy theory and motivic homotopy theory over a field k when G = Gal(L/k) for a finite Galois extension L/k. When k is real-closed with algebraic closure L = k[i] we prove that stable C₂-equivariant homotopy theory embeds fully faithfully into the stable motivic homotopy category over k after $\eta$-completion.

Joint with Jeremiah Heller

Trans. Amer. Math. Soc. 368 (2016), no. 11, 8047-8077

We use the motivic Adams-Novikov spectral sequence and arithmetic fracture to compute the first homotopy group of the motivic sphere spectrum over a field with cohomological dimension less than 3. The answer is in terms of Milnor and Hermitian K-theory and supports a conjecture of Morel.

Joint with Paul Arne Østvær

Advances in Mathematics 265 (2014) 97-131

We perform computations in the spectrum Q(5). We also revisit and extend some computations in Q(3) studied by Mahowald and Rezk. Detection of divided beta elements is studied for each of these.

Joint with Mark Behrens

Algebraic & Geometric Topology 16 (2016) 2459-2534

We use the motivic Adams spectral sequence and a local-to-global framework to compute the homotopy of motivic truncated Brown-Peterson spectra over the rationals.

Joint with Paul Arne Østvær

Geometry & Topology 17 (2013) 1671-1706

We introduce equivariant stable motivic homotopy theory and use it to solve Thomason's homotopy limit problem for Hermitian K-theory.

Joint with Po Hu and Igor Kriz

Advances in Mathematics 228 (2011), no. 1, 434-480

We use the motivic Adams spectral sequence to compute the homotopy of motivic truncated Brown-Peterson spectra over non-dyadic p-adic fields. As an unexpected consequence, we deduce that the slice spectral sequence for motivic BP collapses.

Journal of K-theory 7 (2011), no. 3, 597-618

At the prime 2, over a characteristic 0 field k, the motivic Adams spectral sequence for a finite type cell spectrum X converges to the homotopy of the 2-completion of X whenever cd₂k(i) is finite. Odd-primary results are also described.

Joint with Po Hu and Igor Kriz

Journal of K-theory 7 (2011), no. 3, 573-596

We study the motivic Adams and Adams-Novikov spectral sequences and the motivic J-homomorphism over algebraically closed characteristic 0 fields.

Joint with Po Hu and Igor Kriz

Journal of K-theory 7 (2011), no.1, 55-89

Covers the material in Motivic invariants of p-adic fields and uses those results to study the motivic alpha family in the motivic Adams-Novikov spectral sequence over p-adic fields.

Ph.D. thesis, University of Michigan

This paper explores the Tambara functor structure of the trace ideal of a Galois extension. In the case of a (pro-)cyclic extension, we are able to explicitly determine the generators of the ideal. Furthermore, we show that the absolute trace ideal of a cyclic group is strongly principal when viewed as an ideal of the Burnside Tambara Functor. Applying our results, we calculate the trace ideal for extensions of finite fields. The appendix determines a formula for the norm of a quadratic form over an arbitrary finite extension of a finite field.

Maxine Calle (Reed '20) and Sam Ginnett (Reed '21), with an appendix by Harry Chen (Reed '22) and Xinling Chen (Reed '21)

Submitted, 25pp.

Project Project is a Reed College summer research Project dedicated to Projecting mathematical ideas into the visual realm. The team consists of Henry Blanchette, Cameron Fish, Chris Henn, Kyle Ormsby, Lana Tollas, and Jalan Ziyad. Students wrote blog posts on what they made, how they made it, and the math underlying it all. You can view many of the models in-person in my office.

Ricardo Rojas-Echeneique produces necessary and sufficient conditions for injectivity (resp. surjectivity) of Dress's trace map from the Burnside ring of the Galois group $G$ of a finite Galois extension $L/k$ to the Grothendieck-Witt ring of $k$. These results imply strong necessary conditions for faithfulness (resp. fullness) of the functor $c_{L/k}^*$ introduced by Heller and me in our paper Galois equivariance and stable motivic homotopy theory.

Ricardo Rojas-Echenique, Reed College '17

J. Pure Appl. Algebra 220 (2016), no. 12, 3816-3820

Given a field $F$ of characteristic different from $2$, Riley Thornton determines the structure of the homogeneous Zariski spectrum of the Milnor-Witt $K$-theory of $F$. This space plays a distinguished role in the tensor triangular geometry of the stable motivic homotopy category. Thornton's computation is crucial input to the paper Primes and fields in stable motivic homotopy theory by Heller and myself.

Riley Thornton, Reed College '16

J. Algebra 459 (2016), 376-388