Preprints and publications
- Multiplicative Structure in the Stable Splitting of ΩSLn(C) with Allen Yuan (2017).
The space of based loops in SLn(C), also known as the affine Grassmannian of SLn(C), admits an E2 or fusion product. Work of Mitchell and Richter proves that this based loop space stably splits as an infinite wedge sum. We prove that the Mitchell--Richter splitting is coherently multiplicative, but not E2. Nonetheless, we show that the splitting becomes E2 after base-change to complex cobordism. Our proof of the A∞ splitting involves on the one hand an analysis of the multiplicative properties of Weiss calculus, and on the other a use of Beilinson--Drinfeld Grassmannians to verify a conjecture of Mahowald and Richter. Other results are obtained by explicit, obstruction-theoretic computations.
- Real Orientations of Morava E-theories with Xiaolin Danny Shi (2017). We show that Morava E-theories at the prime 2 are Real oriented and Real Landweber exact. The proof is an application of the Goerss-Hopkins-Miller theorem to algebras with involution. For each height n, we compute the entire homotopy fixed point spectral sequence for En with its C2-action by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these C2-fixed points.
- Nilpotence in En algebras (2017).
Nilpotence in the homotopy of E∞-ring spectra is detected by the classical HZ-Hurewicz homomorphism. Inspired by questions of Mathew, Noel, and Naumann, we investigate the extent to which this criterion holds in the homotopy of En-ring spectra. For all odd primes p and all chromatic heights h, we use the Cohen-Moore-Neisendorfer theorem to construct examples of K(h)-local, E2n-1-algebras with non-nilpotent pn-torsion. We exploit the interaction of the Bousfield-Kuhn functor on odd spheres and Rezk's logarithm to show that our bound is sharp at height 1, and remark on the situation at height 2.
- On the Bousfield classes of H∞-ring spectra (2016).
We prove that any K(n)-acyclic, H∞-ring spectrum is K(n+1)-acyclic, affirming an old conjecture of Mark Hovey.
Appendix to: Brown Peterson cohomology from Morava E-theory by Tobias Barthel and Nathaniel Stapleton (2015).
We provide the proof of a technical lemma about torsion in the Morava E theory of finite abelian groups modulo transfers. The theorems of Barthel and Stapleton then allow conclusions about BP cohomology modulo transfers. Accepted by Compositio Mathematica.
Work from 2013-2018 was supported by an NSF GRFP under Grant DGE-1144152.