I like to think about various topics in homotopy theory and higher algebra.

Here is my CV.

My email address is dwilson@math.harvard.edu.

Here is my CV.

My email address is dwilson@math.harvard.edu.

Here is a list of papers:

A motivic filtration on the topological cyclic
homology of commutative ring spectra. June 2022. With
Jeremy Hahn and Arpon Raksit.

For a prime number $p$ and a $p$-quasisyntomic commutative ring $R$, Bhatt--Morrow--Scholze defined motivic filtrations on the $p$-completions of $\THH(R), \TC^{-}(R), \TP(R),$ and $\TC(R)$, with the associated graded objects for $\TP(R)$ and $\TC(R)$ recovering the prismatic and syntomic cohomology of $R$, respectively. We give an alternate construction of these filtrations that applies also when $R$ is a well-behaved commutative ring spectrum; for example, we can take $R$ to be $\mathbb{S}$, $\mathrm{MU}$, $\mathrm{ku}$, $\mathrm{ko}$, or $\mathrm{tmf}$. We compute the mod $(p,v_1)$ syntomic cohomology of the Adams summand $\ell$ and observe that, when $p \ge 3$, the motivic spectral sequence for $V(1)_*\mathrm{TC}(\ell)$ collapses at the $\mathrm{E}_2$-page.

For a prime number $p$ and a $p$-quasisyntomic commutative ring $R$, Bhatt--Morrow--Scholze defined motivic filtrations on the $p$-completions of $\THH(R), \TC^{-}(R), \TP(R),$ and $\TC(R)$, with the associated graded objects for $\TP(R)$ and $\TC(R)$ recovering the prismatic and syntomic cohomology of $R$, respectively. We give an alternate construction of these filtrations that applies also when $R$ is a well-behaved commutative ring spectrum; for example, we can take $R$ to be $\mathbb{S}$, $\mathrm{MU}$, $\mathrm{ku}$, $\mathrm{ko}$, or $\mathrm{tmf}$. We compute the mod $(p,v_1)$ syntomic cohomology of the Adams summand $\ell$ and observe that, when $p \ge 3$, the motivic spectral sequence for $V(1)_*\mathrm{TC}(\ell)$ collapses at the $\mathrm{E}_2$-page.

Redshift and multiplication for truncated
Brown-Peterson spectra. March 2022. With
Jeremy Hahn.

**Annals of Mathematics.**

We equip BP<n> with an E_{3}-BP-algebra structure, for each prime p and height n. The algebraic K-theory of this E_{3}-ring is of chromatic height exactly n+1.
Specifically, it is an fp-spectrum of fp-type n+1, which can be viewed as a higher height version of the
Lichtenbaum-Quillen conjecture.

We equip BP<n> with an E

On the C_{p}-equivariant dual Steenrod algebra March 2021. With
Krishanu Sankar.

*Proceedings of the American Mathematical Society.*

We compute the C_{p}-equivariant dual Steenrod algebras associated to the constant Mackey functors
__F___{p} and __Z___{(p)}, as __Z___{(p)}-modules. The C_{p} mod p dual Steenrod algebra
is not a direct sum of RO(C_{p})-graded suspensions of __F___{p} when p is odd, in contrast with the classical
and C_{2}-equivariant dual Steenrod algebras.

We compute the C

Odd primary analogs of real orientations. September 2020. With
Jeremy Hahn and Andrew Senger.

*Geometry and Topology.*

We define, in C_{p}-equivariant homotopy theory for p>2, a notion of µ_{p}-orientation analogous to a C_{2}-equivariant Real orientation. The definition hinges on a C_{p}-space CP_{µp}, which we prove to be homologically even in a sense generalizing recent C_{2}-equivariant work on conjugation spaces.
We prove that the height p-1 Morava E-theory is µ_{p}-oriented and that tmf(2) is µ_{3}-oriented. We explain how a single equivariant map S^{2ρ} → Σ^{∞}CP_{µp} completely generates the homotopy of E_{p-1} and tmf(2), expressing a height-shifting phenomenon pervasive in equivariant chromatic homotopy theory.

We define, in C

Real topological Hochschild homology and the Segal conjecture. November 2019. With
Jeremy Hahn.

**Advances in Mathematics.**

We give a new proof, independent of Lin’s theorem, of the Segal conjecture for the cyclic group of order two. The key input is a calculation, as a Hopf algebroid, of the Real topological Hochschild homology of F_2. This determines the E2-page of the descent spectral sequence for the map from NF_2 to F_2, where NF_2 is the C2-equivariant Hill–Hopkins–Ravenel norm of F_2. The E2-page represents a new upper bound on the RO(C_2)-graded homotopy of NF_2, from which the Segal conjecture is an immediate corollary.

We give a new proof, independent of Lin’s theorem, of the Segal conjecture for the cyclic group of order two. The key input is a calculation, as a Hopf algebroid, of the Real topological Hochschild homology of F_2. This determines the E2-page of the descent spectral sequence for the map from NF_2 to F_2, where NF_2 is the C2-equivariant Hill–Hopkins–Ravenel norm of F_2. The E2-page represents a new upper bound on the RO(C_2)-graded homotopy of NF_2, from which the Segal conjecture is an immediate corollary.

Mod 2 power operations revisited. April 2019.

**Algebraic & Geometric Topology.**

In this mostly expository note we take advantage of homotopical and algebraic advances to give a modern account of power operations on the mod 2 homology of commutative ring spectra. The main advance is a quick proof of the Adem relations utilizing the Tate-valued Frobenius as a homotopical incarnation of the total power operation. We also give a streamlined derivation of the action of power operations on the dual Steenrod algebra.

In this mostly expository note we take advantage of homotopical and algebraic advances to give a modern account of power operations on the mod 2 homology of commutative ring spectra. The main advance is a quick proof of the Adem relations utilizing the Tate-valued Frobenius as a homotopical incarnation of the total power operation. We also give a streamlined derivation of the action of power operations on the dual Steenrod algebra.

Eilenberg-MacLane spectra as equivariant Thom spectra.
(
arXiv link). April 2018. With Jeremy Hahn.
**Geometry and Topology.**

We prove that the G-equivariant mod p Eilenberg--MacLane spectrum arises as an equivariant Thom spectrum for any finite, p-power cyclic group G, generalizing a result of Behrens and the second author in the case of the group C_{2}.
We also establish a construction of H__Z___{(p)},
and prove intermediate results that may be of independent interest.
Highlights include an interesting action on quaternionic
projective space, and an analysis of the extent to which the
non-equivariant H**F**_{p} arises as the Thom spectrum
of a more than double loop map.

We prove that the G-equivariant mod p Eilenberg--MacLane spectrum arises as an equivariant Thom spectrum for any finite, p-power cyclic group G, generalizing a result of Behrens and the second author in the case of the group C

A C_{2}-equivariant analog of Mahowald's Thom spectrum theorem.
(arXiv link). Jul. 2017. With Mark Behrens.
**Proceedings of the American Mathematical Society.**

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

C_{2}-Equivariant Homology Operations: Results and Formulas. April 2019.

In this note we state corrected and expanded versions of our previous results on power operations for C_{2}-equivariant Bredon homology
with coefficients in the constant Mackey functor with mod 2 coefficients.
In particular, we give a version of the Adem relations. The proofs
rely on certain results in
equivariant higher algebra which we will supply in a longer
version of this paper.

In this note we state corrected and expanded versions of our previous results on power operations for C

Quotients of even rings. Sep. 2018. With Jeremy Hahn.

We prove that if R is an**E**_{2}-ring with homotopy
concentrated in even degrees, and {x_{j}} is a sequence
of elements in even degrees, then R/(x_{1}, ...) admits
the structure of an **E**_{1}-R-algebra. This
removes an assumption, common in the literature, that
{x_{j}} be a regular sequence.

We prove that if R is an

On categories of slices. Nov. 2017.

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

Power operations for H__F___{2} and
a cellular construction of BP**R**. Nov. 2016.

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. Jul. 2015.

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