Dylan Wilson's Homepage
Assistant Professor of Mathematics
About (CV)
I am an Assistant Professor at WVU. I grew up in (variously) Chicago, IL; Bethesda, MD; and Seattle, WA. I studied mathematics as an undergraduate at the University of Washington and went on to get my Ph.D. in mathematics at Northwestern University. Afterwards I was a postdoctoral researcher at the University of Chicago, followed by an NSF Postdoctoral Fellow and instructor at Harvard University.
My research is in the study of geometric and arithmetically motivated problems (e.g. manifolds and number theory) using the tools of homotopy theory and higher algebra. In other words, I study the qualitative features of (high-dimensional) shapes by constructing and computing certain algebraic measurements of those shapes that are simpler to understand.
As an instructor, I am interested in inclusive pedagogy; in cultivating curiosity, mindfulness and a growth mindset; and in supporting students as best I can to explore past the edges of wherever they currently are.
Papers
- Examples of disk algebras. Feb 2023. With Sanath Devalapurkar, Jeremy Hahn, Tyler Lawson, and Andrew Senger.
- We produce refinements of the known multiplicative structures on the Brown--Peterson spectrum BP, its truncated variants BP<n>, Ravenel's spectra X(n), and evenly graded polynomial rings over the sphere spectrum. Consequently, topological Hochschild homology relative to these rings inherits a circle action.
- 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 S, MU, ku, ko, or tmf. We compute the mod (p,v1) syntomic cohomology of the Adams summand and observe that, when p=3, that the corresponding motivic spectral sequence collapses at the E2-page.
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Redshift and multiplication for truncated
Brown-Peterson spectra. March 2022. With
Jeremy Hahn.
Annals of Mathematics.
- We equip BP<n> with an E3-BP-algebra structure, for each prime p and height n. The algebraic K-theory of this E3-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.
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On the Cp-equivariant dual Steenrod algebra March 2021. With
Krishanu Sankar.
Proceedings of the American Mathematical Society.
- We compute the Cp-equivariant dual Steenrod algebras associated to the constant Mackey functors Fp and Z(p), as Z(p)-modules. The Cp mod p dual Steenrod algebra is not a direct sum of RO(Cp)-graded suspensions of Fp when p is odd, in contrast with the classical and C2-equivariant dual Steenrod algebras.
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Odd primary analogs of real orientations. September 2020. With
Jeremy Hahn and Andrew Senger.
Geometry and Topology.
- We define, in Cp-equivariant homotopy theory for p>2, a notion of µp-orientation analogous to a C2-equivariant Real orientation. The definition hinges on a Cp-space CPµp, which we prove to be homologically even in a sense generalizing recent C2-equivariant work on conjugation spaces. We prove that the height p-1 Morava E-theory is µp-oriented and that tmf(2) is µ3-oriented. We explain how a single equivariant map S2ρ → Σ∞CPµp completely generates the homotopy of Ep-1 and tmf(2), expressing a height-shifting phenomenon pervasive in equivariant chromatic homotopy theory.
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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.
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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.
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Eilenberg-MacLane spectra as equivariant Thom spectra.
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 C2. We also establish a construction of HZ(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 HFp arises as the Thom spectrum of a more than double loop map.
- A C2-equivariant analog of Mahowald's Thom spectrum theorem. Jul. 2017. With Mark Behrens. Proceedings of the American Mathematical Society.
- 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.
- 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 C2-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.
- 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 {xj} is a sequence of elements in even degrees, then R/(x1, ...) admits the structure of an E1-R-algebra. This removes an assumption, common in the literature, that {xj} be a regular sequence.
- 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 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. 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 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.
Teaching
Here is a list of courses I've taught.