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.

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

A C_{2}-equivariant analog of Mahowald's Thom spectrum theorem. With Mark Behrens.
Submitted for publication.

We prove that the C_{2}-equivariant Eilenberg-MacLane
spectrum associated with the constant Mackey functor
__F___{2} is equivalent to
a Thom spectrum over Ω^{ρ}S^{ρ+1}.

We prove that the C

Power operations for H__F___{2} and
a cellular construction of BP**R**. Submitted for publication.

We develop a bit of the theory of power operations for C_{2}-equivariant homology with constant coefficients at
**F**_{2}.
In particular, we construct RO(C_{2})-graded Dyer-Lashof operations
and study their action on an equivariant dual Steenrod algebra.
As an application, we give a cellular construction of BP**R**,
after Priddy.
We study some power operations for ordinary C_{2}-equivariant
homology with coefficients in the constant Mackey functor at **F**_{2}. In addition to a few foundational
results, we calculate the action of these power operations on a C_{2}-equivariant dual Steenrod
algebra. As an application, we give a cellular construction of the C_{2}-spectrum BP**R**
and deduce
its slice tower.

We develop a bit of the theory of power operations for C

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 tmf_{0}(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.

Using the methods of Ando-Hopkins-Rezk, we describe the characteristic series arising from E