Papers & Preprints
Vanishing in stable motivic homotopy sheaves

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.

FM$\Sigma$
arXiv
The stable Galois correspondence for real closed fields

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

ConMath
arXiv
Primes and fields in stable motivic homotopy theory

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.

G&T
arXiv
On the ring of cooperations for 2-primary connective topological modular forms

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

arXiv
Galois equivariance and stable motivic homotopy theory

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

AMS
arXiv
Stable motivivc π1 of low-dimensional fields

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

arXiv
On the homotopy of Q(3) and Q(5) at the prime 2

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

AGT
arXiv
Motivic Brown-Peterson invariants of the rationals

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

G&T
arXiv
The homotopy limit problem for Hermitian K-theory, equivariant motivic homotopy theory and motivic real cobordism

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

PDF

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

JKT
arXiv
Convergence of the motivic Adams spectral sequence

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

JKT
PDF
Remarks on motivic homotopy theory over algebraically closed fields

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

JKT
PDF
Computations in stable motivic homotopy theory

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

Student papers mentored
Title, Description, & Status Paper
Injectivity and surjectivity of the Dress map

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

JPAA
arXiv
The homogeneous spectrum of Milnor-Witt $K$-theory

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

J. Alg
arXiv

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