Title, Description, & Status  Link 

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 C_{2}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), 19 
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 MilnorWitt $K$theory is surjective. Joint with Jeremiah Heller Geom. Topol. 22 (2018), no. 4, 21872218. 
G&T arXiv 
On the ring of cooperations for 2primary connective topological modular forms
We relate three different schemes for understanding tmf∧tmf at the prime 2: boBrownGitler modules, 2variable 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 Gequivariant 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 realclosed with algebraic closure L = k[i] we prove that stable C_{2}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, 80478077 
AMS arXiv 
Stable motivivc π_{1} of lowdimensional fields
We use the motivic AdamsNovikov 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 Ktheory and supports a conjecture of Morel. Joint with Paul Arne Østvær Advances in Mathematics 265 (2014) 97131

Adv 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) 24592534 
AGT arXiv 
Motivic BrownPeterson invariants of the
rationals
We use the motivic Adams spectral sequence and a localtoglobal framework to compute the homotopy of motivic truncated BrownPeterson spectra over the rationals. Joint with Paul Arne Østvær Geometry & Topology 17 (2013) 16711706 
G&T arXiv 
The homotopy limit problem for Hermitian Ktheory,
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 Ktheory. Joint with Po Hu and Igor Kriz Advances in Mathematics 228 (2011), no. 1, 434480 
Adv 
Motivic Invariants of padic fields
We use the motivic Adams spectral sequence to compute the homotopy of motivic truncated BrownPeterson spectra over nondyadic padic fields. As an unexpected consequence, we deduce that the slice spectral sequence for motivic BP collapses. Journal of Ktheory 7 (2011), no. 3, 597618 
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 2completion of X whenever cd_{2}k(i) is finite. Oddprimary results are also described. Joint with Po Hu and Igor Kriz Journal of Ktheory 7 (2011), no. 3, 573596 
JKT 
Remarks on motivic homotopy theory over algebraically
closed fields
We study the motivic Adams and AdamsNovikov spectral sequences and the motivic Jhomomorphism over algebraically closed characteristic 0 fields. Joint with Po Hu and Igor Kriz Journal of Ktheory 7 (2011), no.1, 5589 
JKT 
Computations in stable motivic homotopy theory
Covers the material in Motivic invariants of padic fields and uses those results to study the motivic alpha family in the motivic AdamsNovikov spectral sequence over padic fields. Ph.D. thesis, University of Michigan 
Title, Description, & Status  Paper 

Injectivity and surjectivity of the Dress map
Ricardo RojasEcheneique 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 GrothendieckWitt 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 RojasEchenique, Reed College '17 J. Pure Appl. Algebra 220 (2016), no. 12, 38163820 
JPAA arXiv 
The homogeneous spectrum of MilnorWitt $K$theory
Given a field $F$ of characteristic different from $2$, Riley Thornton determines the structure of the homogeneous Zariski spectrum of the MilnorWitt $K$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), 376388 
J. Alg arXiv 
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