Note: The papers here are more likely to be up to date than the versions on the arXiv.
On categories of slices. Preprint.
In this paper we give an algebraic description of the category of n-slices for an arbitrary group G, in the sense of Hill-Hopkins-Ravenel. Specifically, given a finite group G and an integer n, we construct an explicit G-spectrum W (called an isotropic slice n-sphere) with the following properties: (i) the n-slice of a G-spectrum X is equivalent to the data of a certain quotient of the Mackey functor [W, X] as a module over the endomorphism Green functor [W,W]; (ii) the category of n-slices is equivalent to the full subcategory of right modules over [W,W] for which a certain restriction map is injective. We use this theorem to recover the known results on categories of slices to date, and exhibit the utility of our description in several new examples. We go further and show that the Green functors [W,W] for certain slice n-spheres have a special property (they are geometrically split) which reduces the amount of data necessary to specify a [W,W]-module. This step is purely algebraic and may be of independent interest.
A C2-equivariant analog of Mahowald's Thom spectrum theorem. With Mark Behrens. Submitted for publication.
We prove that the C2-equivariant Eilenberg-MacLane spectrum associated with the constant Mackey functor F2 is equivalent to a Thom spectrum over ΩρSρ+1.
Power operations for HF2 and a cellular construction of BPR. Submitted for publication.
We develop a bit of the theory of power operations for C2-equivariant homology with constant coefficients at F2. In particular, we construct RO(C2)-graded Dyer-Lashof operations and study their action on an equivariant dual Steenrod algebra. As an application, we give a cellular construction of BPR, after Priddy. We study some power operations for ordinary C2-equivariant homology with coefficients in the constant Mackey functor at F2. In addition to a few foundational results, we calculate the action of these power operations on a C2-equivariant dual Steenrod algebra. As an application, we give a cellular construction of the C2-spectrum BPR and deduce its slice tower.
Orientations and Topological Modular Forms with Level Structure. Preprint.
Using the methods of Ando-Hopkins-Rezk, we describe the characteristic series arising from E-genera valued in topological modular forms with level structure. We give examples of such series for tmf0(N) and show that the Ochanine genus comes from an E-ring map. We also show that, away from 6, certain tmf orientations of MString descend to orientations of MSpin.