Hood Chatham
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.
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.
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.
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.
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 $\mathrm{BP}\langle n1 \rangle$ 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 $p1$ Real $\mathit{EO}$ Theory
— Let $p$ be an odd prime and let $EO=E_{p1}^{hC_{p}}$ be the
$C_p$ fixed points of height $p1$ 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
$2k(p1)$ has algebraic $\mathit{EO}$ theory. As an application, we
answer a question posed by Hovey and Ravenel by producing a unital
orientation $\mathit{MY}_{4p4}\to\mathit{EO}$ 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:
BrownPeterson Cohomology of QS^{2n} , and Strickland:
Finite Subgroups of Formal Groups,
Rational Morava E Theory of DS^{0}, 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
BrownPeterson Cohomology of QS^{2n}
and almost all of the relevant part of
Finite Subgroups minus some proofs of
statements. What's missing is the entirety of
Rational Morava ETheory of DS^{0}, which is a fun reasonable paper, and pages 714 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
I am one of the maintainers of
Pyodide, a project to run Python
3.9 code and scientific computing packages in the browser. I wrote some
latex packages:

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.