Hood Chatham

• UCLA Department of Mathematics
• Office MS 6903
• hood@math.ucla.edu

I am an NSF postdoc at UCLA working in algebraic topology. My postdoctoral advisor is Mike Hill. My thesis advisor at MIT was Haynes Miller. I was an undergraduate at Berkeley. My CV is here.

Spectral Sequences

I am curating a collection of spectral sequence diagrams here, I hope they are useful to you. Some of them are pdfs made using my spectralsequences latex package and others are made using an interactive spectral sequence display and editor that I am developing. The source code for the interactive display is on github.

Ext over the Steenrod Algebra

I am working on a minimal resolver which you can find here. It's implemented in Rust and compiled to web assembly. You can find the source code at this gitub repository. They are displayed with my javascipt spectral sequence display. If you clone the repository and install rust, you can run a native binary of the code. It seems to be about 30% faster to run the code natively. Currently it can produce a resolution of an arbitrary finite dimensional or finitely presented Steenrod module over any sub Hopf algebra of the Steenrod algebra using either the Milnor or Adem basis. It also works for the complex motivic Steenrod algebra at p=2. The old C resolver can do unstable resolutions though it doesn't use the correct instability condition at odd primes. I believe that it's the fastest currently available resolver, and is certainly the most convenient. It is possible to save the output and interactively add differentials to it. For example, I made charts of the Adams spectral sequence for S₍₂₎ and the Adams spectral sequence for S₍₃₎ in this way.

The Adams Spectral Sequence for $\tmf_{(2)}$

Dexter Chua has documented the calculation of the Adams spectral sequence for $\tmf_{(2)}$ here. He used the Ext resolver to compute the $E_2$ page and products and to propagate differentials by the Leibniz rule.

Steenrod Calculator

In the process of creating the ext calculator, I had to implement the Adem and Milnor bases for the Steenrod algebra. This simple interface parses an arbitrary arithmetic expression using any mixture of Adem and Milnor monomials, addition, multiplication, and parentheses. It can output both into the Milnor basis and into the Adem basis. For simple use cases itis probably both faster and easier to use in than the Sage steenrod algebra implementation.

Papers

• Wilson Spaces, Snaith Constructions, and Elliptic Orientations joint with Jeremy Hahn and Allen Yuan — We construct a canonical family of even periodic $\mathbb{E}_{\infty}$-ring spectra, with exactly one member of the family for every prime $p$ and chromatic height $n$. At height $1$ our construction is due to Snaith, who built complex $K$-theory from $\mathbb{CP}^{\infty}$. At height $2$ we replace $\mathbb{CP}^{\infty}$ with a $p$-local retract of $\mathrm{BU} \langle 6 \rangle$, producing a new theory that orients elliptic, but not generic, height $2$ Morava $E$-theories. In general our construction exhibits a kind of redshift, whereby $\mathrm{BP}\langle n-1 \rangle$ is used to produce a height $n$ theory. A familiar sequence of Bocksteins, studied by Tamanoi, Ravenel, Wilson, and Yagita, relates the $K(n)$-localization of our height $n$ ring to work of Peterson and Westerland building $E_n^{hS\mathbb{G}^{\pm}}$ from $\mathrm{K}(\mathbb{Z},n+1)$.
• An Orientation Map for Height $p-1$ Real $\mathit{EO}$ Theory — Let $p$ be an odd prime and let $EO=E_{p-1}^{hC_{p}}$ be the $C_p$ fixed points of height $p-1$ Morava $E$ theory. We say that a spectrum $X$ has algebraic $\mathit{EO}$ theory if the splitting of $K_*(X)$ as an $K_*[Cp]$-module lifts to a topological splitting of $\mathit{EO}\wedge X$. We develop criteria to show that a spectrum has algebraic $\mathit{EO}$ theory, in particular showing that any connective spectrum with mod $p$ homology concentrated in degrees $2k(p-1)$ has algebraic $\mathit{EO}$ theory. As an application, we answer a question posed by Hovey and Ravenel by producing a unital orientation $\mathit{MY}_{4p-4}\to\mathit{EO}$ analogous to the $\mathit{MSU}$ orientation of $\mathit{KO}$ at $p=2$. We prove as corollaries that the $p$th tensor power of any virtual dimension zero vector bundle is $\mathit{EO}$-oriented and that $p$ times any vector bundle is $\mathit{EO}$-oriented.
• Thom Complexes and the Spectrum $\tmf\,$ — Following Mahowald's arugment that $\mathit{bo}$ and $\mathit{bu}$ are not $E_1$ Thom spectra, we prove that $\mathit{tmf}$ is not an $E_1$-Thom spectrum.

Talk Notes

• Lubin Tate Spectra and the Goerss Hopkins Miller Theorem, notes from a talk I gave at Oberwolfach and at Juvitop in Spring 2018 about Jacob Lurie's proof of the Goerss Hopkins Miller theorem.
• Goodwillie differentials and Hopf invariants, notes from a talk I gave for Juvitop in Fall 2017 about chapter four of The Goodwillie tower and the EHP sequence. It contains diagrams illustrating the generalized geometric boundary theorem, which I have found useful because the statement itself is a bit hard to read.
• The Goodwillie Tower of the Identity, notes from a talk I gave for Juvitop in Fall 2017 about Arone Mahowald's paper.

• Strickland's Theorem, notes from a talk I gave for Juvitop in Fall 2016 on Strickland's theorem that formal spec of the ring of additive degree n operations on Morava E theory is the scheme of degree n subgroups of the formal group. These are incomplete attempt at a completely self contained source on the parts of the following four papers that are necessary for Strickland's result: Kashiwabara: Brown-Peterson Cohomology of QS²ⁿ , and Strickland: Finite Subgroups of Formal Groups, Rational Morava E Theory of DS⁰, and the main paper The Morava E theory of Symmetric Groups.

  My notes contain all of the hardest parts of these four papers. What is present is the main theorem, all of the goodness arguments, all of the relevant part of Brown-Peterson Cohomology of QS²ⁿ and almost all of the relevant part of Finite Subgroups minus some proofs of statements. What's missing is the entirety of Rational Morava E-Theory of DS⁰, which is a fun reasonable paper, and pages 7-14 of the main paper, which are very dry and technical. Even with these omissions and the mistakes that are certainly present, I think these notes are a very good resource for anyone trying to learn about this theorem.

• The Positive Complete Model Structure and Why We Need It, notes from a talk I gave at Talbot on April 5th, 2016 on deriving the symmetric power functor. Thanks to Eva for providing a livetexed skeleton. This appears as part of the complete Talbot 2016 proceedings here.
• Algebraic Structures in Equivariant Homotopy Theory, notes from a talk I gave for Juvitop in Spring 2016 on the Lawvere theories of Mackey and Tambara functors.
• Tannakian Categories, notes for a talk I gave for Gonçalo Tabuada's class on motivic homotopy theory in Spring 2016

Programming

• spectralsequences: Print spectral sequence diagrams using pgf/tikz. The development repository is here.
• longdivision: Print solutions to long division problems with divisors up to 9 digits long and dividends only limited by page size. Finds repeating decimal pattern if the first repeated remainder occurs before it runs out of page space.
• tikzcdintertext: Defines a command \intertext inside of tikzcd which acts like the \intertext command from amsmath. Here's an example file and here's the output.