I like to think about various topics in homotopy theory and higher algebra.
My email address is dwilson@math.harvard.edu.
Here is a list of papers:
On the Cp-equivariant dual Steenrod algebra March 2021. With
Krishanu Sankar.
We compute the C
p-equivariant dual Steenrod algebras associated to the constant Mackey functors
Fp 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
Fp when p is odd, in contrast with the classical
and C
2-equivariant dual Steenrod algebras.
Redshift and multiplication for truncated
Brown-Peterson spectra. December 2020. With
Jeremy Hahn.
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.
Odd primary analogs of real orientations. September 2020. With
Jeremy Hahn and Andrew Senger.
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.
Real topological Hochschild homology and the Segal conjecture. November 2019. With
Jeremy Hahn.
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.
C2-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.
Mod 2 power operations revisited. April 2019.
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.
Quotients of even rings. Sep. 2018. With Jeremy Hahn.
We prove that if R is an
E2-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
E1-R-algebra. This
removes an assumption, common in the literature, that
{x
j} be a regular sequence.
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
Fp arises as the Thom spectrum
of a more than double loop map.
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.
Older papers, superseded or awaiting revision:
Power operations for HF2 and
a cellular construction of BPR. Nov. 2016.
We develop a bit of the theory of power operations
for C
2-equivariant homology with constant coefficients at
F2.
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
F2. 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.
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.