Preprints and publications
 Multiplicative Structure in the Stable Splitting of ΩSL_{n}(C) with Allen Yuan (2017).
The space of based loops in SL_{n}(C), also known as the affine Grassmannian of SL_{n}(C), admits an E_{2} 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 MitchellRichter splitting is coherently multiplicative, but not E_{2}. Nonetheless, we show that the splitting becomes E_{2} after basechange 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 BeilinsonDrinfeld Grassmannians to verify a conjecture of Mahowald and Richter. Other results are obtained by explicit, obstructiontheoretic computations.
 Real Orientations of Morava Etheories with Xiaolin Danny Shi (2017). We show that Morava Etheories at the prime 2 are Real oriented and Real Landweber exact. The proof is an application of the GoerssHopkinsMiller theorem to algebras with involution. For each height n, we compute the entire homotopy fixed point spectral sequence for E_{n} with its C_{2}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 C_{2}fixed points.
 Nilpotence in E_{n} algebras (2017).
Nilpotence in the homotopy of E_{∞}ring spectra is detected by the classical HZHurewicz homomorphism. Inspired by questions of Mathew, Noel, and Naumann, we investigate the extent to which this criterion holds in the homotopy of E_{n}ring spectra. For all odd primes p and all chromatic heights h, we use the CohenMooreNeisendorfer theorem to construct examples of K(h)local, E_{2n1}algebras with nonnilpotent p^{n}torsion. We exploit the interaction of the BousfieldKuhn 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 Etheory 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 20132018 was supported by an NSF GRFP under Grant DGE1144152.
