Preprints
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The invariant cell number property for E4-rings (joint with Chedalavada)
[arxiv link]
For any non-zero E4-ring R, we prove that an n-fold direct sum of suspensions of R cannot admit
an exhaustive filtration by fewer than n cells. We do this by showing that certain integral representations
of the symmetric group lift to Schur functors on additively E3-monoidal ∞-categories.
We then reduce the statement to a vanishing fact for rational Schur functors due to Deligne.
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Nilpotence phenomena over E∞-rings in positive characteristic
[arxiv link] (submitted)
We study the behavior of nilpotent classes in E∞-rings over finite fields.
In characteristic p=2, we show that any nilpotent class can be killed by a nilpotence detecting map of E∞-rings.
As a consequence, we prove the existence of "residue fields", namely that any non-zero E∞-ring over F2 admits a map
to a separably closed graded field.
Conversely, for any odd prime p, we construct non-zero E∞-rings over Fp which admit nilpotent classes that cannot be
killed by any non-zero map of E∞-rings, a phenomenon that does not occur classically or at any finite height.
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On the K-Theory of algebraic tori
(joint with Bai, Carmeli
and Juran; [img])
[arxiv link] (submitted)
Given an algebraic torus T over a field F, its lattice of characters Λ(T) gives rise to a topological torus BΛ(T) with a continuous action
of the absolute Galois group G. We construct a natural equivalence between the algebraic K-theory of T
and the equivariant homology of BΛ(T) with coefficients in the G-equivariant K-theory of F.
We obtain this equivalence by analyzing the KGL-motive of T in the stable motivic category SH(F) of Voevodsky and Morel.
We construct a natural comparison map as a special case of a "motivic Fourier transform" and prove that it is an equivalence by using a
motivic Eilenberg–Moore formula for classifying spaces of tori.
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On the deformation theory of E∞-coalgebras
[updated version]
[arxiv link] (submitted)
We show that, for any prime p and any space X, the Fp-chains of X admit a unique and functorial lift
to an E∞-coalgebra in the ∞-category of p-complete connective spectra, given by the p-completed suspension spectrum of X.
We do this by introducing a notion of formally étale coalgebras and prove that the coalgebraic right adjoint to base change
along the map Sp -> Fp is fully faithful when restricted to formally étale coalgebras.
We conclude by showing that the F_p-chains are formally étale, by reducing to the case of an Eilenberg--MacLane space
X=K(Fp,n) and using an explicit presentation of the Fp-chains via power operations.
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