This webpage is a place for me to organize my papers into projects and
to provide a bit of explanation for them, to clarify how my individual
publications fit into continuing broader projects with overarching
goals. Most of these papers are also available on the preprint arXiv
and, for the ones that have been published already, on the journal
websites as well. I haven't marked the publication status of each paper
on this page, because that isn't what this page is for. If you want to
know whether a given paper is published, and in what journal, a web
search will tell you that. Alternatively, you can
consult ORCID, which is usually pretty uptodate on published papers.
Versions on this page may be more current
(with corrections of minor typos which don't warrant a full arXiv
update) than the arXiv versions.

"Complex multiplication and homotopy groups of spheres" project, with subtopics:

Other papers in pure mathematics

Papers on data analysis and fMRI

Notes
"Complex multiplication and homotopy groups of spheres" project.
This is the main project I work on. The basic theme is to describe the
orders of the stable homotopy groups of Bousfieldlocalized finite
spectra in terms of special values of Lfunctions and zetafunctions
that arise in number theory. The broad motivation for such a project is
that it is generally extremely difficult to even
describe the
v_nperiodic families in the homotopy groups of spheres for n>1, so
when you do difficult computations in this area, even the
statements
of the theorems (let alone their proofs!) are almost always
frustratingly long and difficult to parse. Sometimes you can get a
simpler and more satisfying statement by describing the orders of the
relevant homotopy groups in terms of something else that's also
difficult to understand but at least wellstudied elsewhere in
mathematicsin particular, special values of some Lfunction or
zetafunction.
In this project I have often found it useful it to use formal groups
with complex multiplication, that is, "formal modules." For a number
field K, some of the flat cohomology groups of the moduli stack of
formal O_Kmodules have orders describable in terms of the Dedekind
zetafunction of K (e.g. see the "Structure and cohomology..." paper,
below). Formal modules also expedite some "heightshifting" tricks,
since if K is a degree d extension of the padic rationals, then the
automorphism group of an O_Kheight n/d formal O_Kmodule naturally
embeds into the height n full Morava stabilizer group as a rather large
profinite subgroup. This is exploited in various ways in the papers
below, particularly in "Moduli of formal Amodules under change of A"
and "Height four formal groups..." and "The cohomology of the height
four...".
The project is far from finished, but so far I have been able to make
progress in several directions, which you can read about in the papers
below. I have also been (slowly) working on a book on this material
which is intended as a companion piece to Ravenel's book "Complex
cobordism and stable homotopy groups of spheres."
The papers in this project fit into several threads, which interact with each other:
 papers on complexanalytic Lfunctions (e.g. automorphic Lfunctions) associated to finite CWcomplexes,
 papers on Iwasawa theory and padic Lfunctions associated to finite CWcomplexes,
 papers on moduli of formal modules and their applications, e.g. via "heightshifting,"
 and papers which carry out calculations in stable homotopy theory which are used to prove results in any of the above topics.
Papers on complexanalytic Lfunctions associated to finite CWcomplexes:

Denominators of special values of zetafunctions count KUlocal homotopy groups of mod p Moore spectra.
The first circulated versions of this paper are from 2016, although I
revised it more than once before finally sending it to a journal, and it
didn't wind up in print until 2023.
In this note, for each odd prime p, I show that the orders
of the KUlocal homotopy groups of the mod p Moore spectrum are equal to
denominators of special values of certain quotients of Dedekind
zetafunctions of totally real number fields. I use this and Colmez's
padic class number formula to give a cute topological proof of the
Leopoldt conjecture for those number fields, by showing that it is a
consequence of periodicity properties of KUlocal stable homotopy
groups.

KUlocal zetafunctions of finite CWcomplexes, 2023.
This paper extends the main result "Denominators of special
values...mod p Moore spectra" by defining a KUlocal zetafunction for
any finite CWcomplex with cohomology concentrated in even degrees,
showing it has analytic continuation to the complex plane, showing it is
equal to a product of Tate twists of Lfunctions of characters, and
showing that its special values at negative integers have denominators
equal to the orders of the KUlocal stable homotopy groups of X, up to
powers of certain primes.

elladic topological JacquetLanglands duality, with an appendix jointly authored with Matthias Strauch, 2023.
It is an old problem in stable homotopy theory to find some
concrete benefit to the appearance of the LubinTate spaces in both
computational homotopy theory (since the cohomology of the Morava
stabilizer group scheme with coefficients in the global sections on
LubinTate space is the input for a descent spectral sequence converging
to the homotopy groups of the K(n)local sphere) and in Carayol's
approach to the construction of Langlands and JacquetLanglands
correspondences. Carayol essentially conjectured that the elladic
vanishing cycle cohomology of the LubinTate tower would realize the
Langlands and JacquetLanglands correspondences in a certain precise
way. This conjecture was later proven by Henniart and by HarrisTaylor.
An old sticking point, for making any fruitful connection to homotopy
theory: SchwanzlVogtWaldhausen showed that the realization of the base
of the LubinTate tower by an E_\inftyring spectrum can't possibly
extend to a realization of the higher stages in the LubinTate tower.
In the appendix of this paper, Matthias Strauch and I
describe our workaround for that old problem: we embed the LubinTate
tower into a larger tower of deformation spaces, which we call
"degenerating LubinTate spaces." We use HopkinsKuhnRavenel to
construct a spectral realization of the degenerating LubinTate spaces,
and we show that the relevant vanishing cycle cohomology of the
classical LubinTate spaces splits off as a summand in the cohomology of
the degenerating LubinTate spaces.
In the main body of the paper, I build some theory of
nearby cycles and vanishing cycles for spectral sheaves, and I use it
together with the spectral realization of the degenerating LubinTate
tower to construct "elladic JacquetLanglands cohomology". This is a
generalized cohomology theory which realizes the elladic
JacquetLanglands and elladic Langlands correspondences for Q_p in its
rational stable homotopy groups. I use it to build an "elladic
topological JacquetLanglands (TJL) dual" of any spectrum with respect
to any onedimensional formal group G over a perfect field of
characteristic p. This dual has the property that the supercuspidal
representations of GL_n(Q_p) occuring in its stable homotopy groups are
the elladic JacquetLanglands duals of representations of Aut(G)
occurring in the Morava Etheory E(G)_*(X) of X. Finally, if G has
height 1, I show that the product of the automorphic Lfactors
associated to the GL_1representations occuring in the TJL dual of X are
precisely the "provisional KUlocal zeta function of X" as defined in
the "KUlocal zetafunctions of finite CWcomplexes" paper, above.
Consequently this Euler product analytically continues to a meromorphic
function on the complex plane, and its special values in the left
halfplane recover the orders of the KUlocal stable homotopy groups of
X, away from 2, irregular primes, and primes at which the homology of X
has nontrivial torsion.
Papers on Iwasawa theory and padic Lfunctions associated to finite CWcomplexes:

Iwasawa invariants of finite spectra, joint with Austin Maison, 2024.
In this paper, Austin Maison and I calculate the Iwasawa
invariants of the Iwasawa modules you get from the padically complete
complex Ktheory of finite CWcomplexes, for an odd prime p. The
resulting Iwasawa invariants are topologically meaningful: for example,
we show that the Iwasawa lambdainvariant measures the asymptotic growth
rate of the K(1)local homotopy groups of the CWcomplex. The
characteristic polynomials yield algebraic padic Lfunctions of finite
CWcomplexes, and we prove a "weak form of the Iwasawa main conjecture"
in this setting, which asserts that the special values of this algebraic
padic Lfunction at negative integers have padic valuations which
agree, up to sign, with the padic valuations of the orders of the
K(1)local homotopy groups of the CWcomplex. In many ways this paper
does for padic Lfunctions what the "KUlocal zetafunctions of finite
CWcomplexes" paper does for complexanalytic zetafunctions.
Papers on moduli of formal modules and their applications:

Structure and cohomology of moduli of formal modules, originally 2015, although rewritten almost entirely in 2023.
This paper proves structural results about the classifying
ring L^A of formal Amodules and about the moduli stack M_{fmA} of
formal Amodules. I use these structural results to aid in explicit
calculations of flat cohomology groups of M_{fmA}^{2buds}, the moduli
stack of formal Amodule 2buds. For example, a generator of the group
H^1_{fl}(M_{fm Z}; omega), which also generates (via the AdamsNovikov
spectral sequence) the first stable homotopy group of spheres, also
yields a generator of the Amodule H^1_{fl}(M_{fmA}^{2buds}; omega) for
any torsionfree Noetherian commutative ring A. I show that the order
of the Amodules H^1_{fl}(M_{fmA}^{2buds}; omega) and
H^2_{fl}(M_{fmA}^{2buds}; omega tensor omega) are each equal to
2^{N_1}, where N_1 is the leading coefficient in the 2local
zetafunction of Spec A.
In this paper I also prove that the cohomology of
M_{fmA}^{2buds} is closely connected to the deltainvariant and
syzygetic ideals studied in commutative algebra:
H^0_{fl}(M_{fmA}^{2buds}; omega tensor omega) is the deltainvariant of
the largest ideal of A which is in the kernel of every ring
homomorphism from A to the field with two elements, and consequently
H^0_{fl}(M_{fmA}^{2buds}; omega tensor omega) vanishes if and only if A
is a ring in which that ideal is syzygetic.

Computation of the classifying ring of formal modules, originally 2015, although I revised it in 2023 with much stronger results.
This paper explicitly computes the classifying ring L^A of
formal Amodules, for A a Dedekind domain of characteristic zero.
Lazard, Drinfeld, and Hazewinkel computed the ring L^A in the case of A a
field or a Dedekind domain of class number one; in either case, L^A is a
polynomial algebra over A. Hazewinkel described a case (specifically,
the case of A the ring of integers in Q with a fourth root of 18
adjoined) where L^A could not possibly be a polynomial algebra, but
Hazewinkel did not compute that ring L^A. It seems that, before this
paper, L^A had never been computed in any situation in which it failed
to be polynomial.
In this paper I obtain an explicit presentation for L^A for
Hazewinkel's ring A, as a corollary of a more general theorem: I show
that, for all Dedekind domains A of characteristic zero, L^A is a
symmetric algebra on a certain projective Amodule. The proof of this
fact is a fun one: it involves producing a homology theory which
measures the failure of a certain comparison map between L^A and a
symmetric algebra to be an isomorphism, and then using a comparison to
Hochschild homology and a Hochschild calculation due to Pirashvili in
order to show vanishing of this homology theory in the relevant degrees.
This leads to explicit presentations for L^A. It also leads to
qualitative results about the moduli theory of formal Amodules, namely,
any formal Amodule over R/I lifts to a formal Amodule over R, and any
formal Amodule nbud extends to a formal Amodule (n+1)bud.

Moduli of formal Amodules under change of A,
2016. (The earliest version of this paper was one of the first things I
wrote, but in 2016 I went back and rewrote the paper from scratch, with
many new results.)
This one sets up the machinery you need in order to
explicitly compute where classes in the cohomology of the height n
Morava stabilizer group are sent, under the restriction map in
cohomology, into the cohomology of the group of automorphisms of a
height n formal group which commute with complex multiplication by A,
where A is the ring of integers in a field extension of the padic
rationals of degree dividing n. This is a pretty useful thing to be able
to do: in the paper I show that, using some local class field theory,
this gets you an action of the Galois group Gal(K^ab/K^nr) on the
Morava/LubinTate spectrum E_n, for each degree n field extension K of
Q_p. The paper also computes the map from the K(2)local homotopy groups
of the SmithToda complex V(1) to the (automatically K(2)local)
homotopy groups of the homotopy fixedpoints of that Galois action on
E_2, for each of the quadratic extensions K/Q_p, for p>3. Another
point of view: this construction is getting you a map from K(n)local
homotopy into the Galois cohomology of K^ab/K^nr for each degree n field
extension K/Q_p. Here K^ab is the maximal abelian extension of K and
K^nr is the maximal unramified extension of K.
This map detects a lot: in the paper we show that, in the
n=2 case, for each element in the K(2)local homotopy of V(1), either
that element or its Poincare dual (exactly one or the other, not both)
maps nontrivially into the Galois cohomology of K^ab/K^nr for some
quadratic K/Q_p. For example, the element zeta_2 from the chromatic
splitting conjecture maps nontrivially into the Galois cohomology of
K^ab/K^nr for K/Q_p unramified, while all the other elements in the
K(2)local homotopy of V(1) map nontrivially into the Galois cohomology
of K^ab/K^nr for K/Q_p totally ramified. (I don't mention it in this
paper, but this phenomenon does appear to happen more generally than in
just the p > 3 quadratic cases; it is more difficult to describe the
phenomenon, however, in the p=3, n=2 and p=5, n=4 cases, where it also
seems to be occuring, since the Morava stabilizer group at those heights
and primes has infinite cohomological dimension and so its cohomology
does not have the "easy" Poincare duality.)

Height four formal groups with quadratic complex multiplication, 2016.
In this paper you can find the more technical (and
powerful) machinery for computing cohomology of automorphism groups of
formal groups with complex multiplication. In Ravenel's "green book" he
sets up a filtration on the Morava stabilizer algebras so that their
associated graded Hopf algebras are (dual to) primitively generated Hopf
algebras, so that the methods of MilnorMoore and Peter May's thesis
can be used to compute the cohomology of strict automorphism groups of
formal groups (since the Morava stabilizer algebras are the continuous
linear duals of the group rings of those strict automorphism groups).
This paper does the same thing for formal groups with complex
multiplication, reducing the problem of computing the cohomology of
their automorphism groups to the problem of computing the cohomology of
solvable Lie algebras using a ChevalleyEilenberg resolution, and then
running certain May spectral sequences.
Then in this paper I carry out the computations to compute
the cohomology of the group G of the strict automorphisms of a height
four formal group which commute with complex multiplication by
Z_p[sqrt(p)], for primes p > 5. This group turns out to have
cohomological dimension 8 and its cohomology has rank 80. (That
computation also appears in the draft version of "The cohomology of the
height four Morava stabilizer group at large primes," below; when that
paper is ready for journal submission, I'll trim that computation in it,
with the idea that "Height four formal groups with quadratic complex
multiplication" is the paper with the authoritative version. This also
tells you one reason why the cohomology of this particular group G
matters: it is a stepping stone on the way to computing the cohomology
of the height four Morava stabilizer group.)
Finally, I show that the automorphism group of a formal group with complex multiplication is a closed
subgroup of the Morava stabilizer group, so we can use the work of
DevinatzHopkins to take the homotopy fixedpoints of the action of this
closed subgroup on a LubinTate/Morava Etheory spectrum. In this paper
I carry out the computation to get the V(3)homotopy of the homotopy
fixed points of the group G, above, acting on E_4, for primes p > 5
(since otherwise the SmithToda complex V(3) does not exist). The result
is 2(p^2  1)periodic, and interpolates between the K(4)local
homotopy groups of V(3) (which are quite complicated, and 2(p^4 
1)periodic) and the LubinTate fixed points arising from Galois
cohomology of totally ramified quartic extensions of Q_p (which are
quite simple, and 2(p  1)periodic), as in the paper "Moduli of formal
Amodules under change of A," above.

Ravenel's algebraic extensions of the sphere spectrum do not exist, 2015.
In a 1983 paper, Ravenel asked whether there exists a
spectrum S^A such that the complex bordism of S^A is isomorphic to L^A
as a module over the coefficient ring MU_* of complex bordism. Here A is
the ring of integers in a finite extension of the padic rationals.
Such a spectrum S^A would then be an "algebraic extension of the sphere
spectrum." In the base case, when A is the ring of padic integers, S^A
exists: it is simply the pcomplete sphere spectrum.Some later work was
done on this problem: A. Pearlman computed the Morava Ktheories of the
spectra S^A, assuming they exist, and T. Lawson showed that, if S^A
exists, then it cannot be a A_pring spectrum, that is, a spectrum with a
ring structure that is associative up to pthorder homotopies. Lawson
also showed that S^A does not exist if A is the ring of integers in the
2adic rationals with a primitive fourth root of unity adjoined, but in
all other cases, Ravenel's original problem remained open.
This paper settles Ravenel's question: S^A does not exist
except in the base case, when A is the ring of padic integers. The
natural global and ptypical analogues of Ravenel's question are also
stated and then shown to have negative answers. These all ultimately
follow from the paper's main technical result, a topological
nonrealizability theorem for modules over the coefficient ring BP_* of
BrownPeterson homology, which implies that the only spectra X such that
BP_*(X) is a V^Amodule are extensions of rational spectra by dissonant
spectra. Here V^A is the classifying ring of Atypical formal
Amodules.
While formal Amodules are not topologically realizable in
the way that Ravenel asked about, they are still very useful for making
computations in topology: see the papers below, where I use them to
compute the homotopy groups of the K(4)local SmithToda complex V(3).

Ravenel's Global Conjecture is true, 2015.
Let A be a (local or global) number ring. In a 1983 paper,
Ravenel made a conjecture, the "Global Conjecture," describing the
orders of the Ext^1 groups of the classifying Hopf algebroid of formal
Amodules, or equivalently, the orders of the flat cohomology groups H^1
of the moduli stack of formal Amodules, in terms of numbers
generalizing Adams's numbers h(f,t) from "On the groups J(X) II." In
this paper I get a proof of the Global Conjecture by a combination of
spectral sequence methods and a kind of Hasse principle for ideals
satisfying conditions generalizing the characterizing properties of
ideals generated by Adams' numbers.
Some interesting things come out of this which go beyond
what Ravenel originally conjectured. In this paper we see that certain
classes of Hecke Lfunctions can be recovered (via their Euler products)
from the flat cohomology group H^1 of the moduli stack of formal
modules. For example, if K/Q and L/Q are finite Galois extensions with
rings of integers A and B respectively, and if we suppose that 2
ramifies in both A and B and that [K : Q] and [L : Q] are odd primes,
then the flat cohomology group H^1 of the moduli stack of formal
Amodules is isomorphic (as a graded abelian group) to the flat
cohomology group of the moduli stack of formal Bmodules if and only if
the Dedekind zetafunction of K is equal to the Dedekind zetafunction
of L. (The assumptions on degree and ramification can be lifted
somewhat; see the paper for details.)
The representations of Galois groups in the cohomology of
LubinTate spaces, and more "globally," Shimura varieties, are often
used to study the zetafunctions associated to those representations.
The moduli stack of formal Amodules is an alternative "globalization"
to a Shimura variety: like a Shimura variety, the formal completion at
each point is a deformation space of a BarsottiTate module (e.g. a
LubinTate space), and in this paper we show that zetafunction data is
also visible in the cohomology of the moduli stack of formal Amodules.
Papers which carry out calculations in stable homotopy theory using or used by papers from the above sections:

The mod p cohomology of the Morava stabilizer group at large primes, joint with my recentlygraduated student Mohammad Behzad Kang, 2024.
In this paper, Kang and I calculate the cohomology of the
full Morava stabilizer group scheme with trivial mod p coefficients, at
all heights, for all primes p>>n. Our answer is that this
cohomology is an exterior algebra on n generators, isomorphic to the mod
p singular cohomology of the unitary group U(n). The existence of such
an isomorphism was an old folklore conjecture in computational stable
homotopy theory. You can consult our paper for various rephrasings of
that cohomology in terms of the extended Morava stabilizer group, the
full Morava stabilizer group (rather than the group scheme), etc. With
these coefficients, what we calculate isn't the cohomology that
yields the whole E_2page for the E(n)local Adams spectral sequence
converging to the E(n)local homotopy groups of the SmithToda complex
V(n1). Instead, what we calculate is only the diagonal line of slope 1
passing through bidegree (0,0), which is then repeated to the left and
the right every 2(p^n1) stems in the E_2page of the spectral sequence.
Our approach involves some new ideas. We produce a family
of DGAs over the affine line A^1 over F_p, smooth except at 0. The
singular fiber (i.e., the fiber at 0) is isomorphic to the
ChevalleyEilenberg DGA of Ravenel's Lie algebra model L(n,n) for the
cohomology of the Morava stabilizer group. The smooth fibers are each
ChevalleyEilenberg DGAs of a reductive Lie algebra isomorphic to
gl_n(F_p). We show there is an essentiallyunique C_nequivariant
multiplicative differentialgraded connection on the resulting bundle of
DGAs over A^1  {0}. From parallel transport in this connection, you
get a PicardLefschetz (i.e., monodromy) operator T on each of the
smooth fibers. The most technically elaborate new tool in the paper is a
derived analogue of the local invariant cycles theorem from algebraic
geometry which lets us compare the Tfixed points of the cohomology of
the smooth fibers to the cohomology of the singular fiber.
There are a few other new ideas in the paper too, other
stuff we wind up needing in order to prove the main theorem, like some
useful material on constructing small models (smaller than the
ChevalleyEilenberg DGA) for the cohomology of reductive Lie algebras.

Ravenel's May spectral sequence collapses immediately at large primes, 2023.
Fix a positive integer n. In this paper I show that, for
p>>n, Ravenel's May spectral sequence converging to the mod p
cohomology of the height n strict Morava stabilizer group scheme
collapses immediately. Consequently, at such primes, the mod p
cohomology of the strict Morava stabilizer group scheme is the
cohomology of a finitedimensional solvable Lie algebra. That means the
cohomology is computable algorithmically, since there's a "brute force"
algorithm for calculating cohomology of finitedimensional Lie algebras
(just calculate the cohomology of the ChevalleyEilenberg complex).
One consequence is the existence, for a prime p>>n
and any E(n1)acyclic finite CWcomplex X, of a spectral sequence whose
input is Lie algebra cohomology tensored with E(n)homology of X, and
which converges to the input for the E(n)Adams spectral sequence for X
(which itself converges to the E(n)local stable homotopy groups of X).
The most important ingredient in the proofs in this paper
is a construction of simultaneous integral deformations of Morava
stabilizer algebras and various related algebraic objects. For example,
given a positive integer n, in this paper I construct a DGA over the
integers which specializes, at each prime p, to the cobar complex of the
pprimary height n Morava stabilizer algebra. Consequently this
integral gadget "knows" the mod p cohomology of the height n Morava
stabilizer group for every prime p. I have found these integral
gadgets useful for some applications beyond from those described in this
paper, and I hope to return to them in later writing.

The cohomology of the height four Morava stabilizer group at large primes, 2016.
This is, for the moment, an announcement and not a finished
paper; I will update the version of the document available here once I
have a complete writeup of the results. Despite this, the main results
are stated and the main tools are developed; the only content missing
from this document is the computation of certain spectral sequence
differentials which I worked out using a computer, and which I am still
figuring out how to explain and typeset concisely and intelligibly in a
paper.
This paper features the computation of the mod p cohomology
of the height 4 Morava stabilizer group, that is, the automorphism
group of a pheight 4 formal group law over the field with p elements,
at primes p > 5. This uses some new "heightshifting" techniques: I
construct a sequence of spectral sequences whose input is the cohomology
of the height n Morava stabilizer group, and whose output is the
cohomology of the automorphism group of an Aheight n formal Amodule,
where A is the ring of integers in a quadratic extension of the padic
rationals; and then we construct a sequence of spectral sequences whose
input is the cohomology of an Aheight n formal Amodule, and whose
output is the cohomology of the height 2n Morava stabilizer group. This
generalizes, by replacing the quadratic extension with an arbitrary
extension K/Q_p and replacing 2n with [K : Q_p] times n, but it is the
quadratic case that this paper is primarily concerned with, since we
actually then run the spectral sequences in the case n = 2 and p > 5.
In the end, the rank (i.e., vector space dimension over F_p) of the
cohomology of the height 4 Morava stabilizer group at large primes is
3,440. (Compare this to 152 for height 3, 12 for height 2, and 2 for
height 1.) These ranks fit into a pattern, given in the paper, which
suggests a conjecture describing the rank of the cohomology of the
height n Morava stabilizer group at large primes, for all n. As far as I
know, this is the first attempt at giving a conjectural description of
the ranks of these cohomology groups for all heights.
There is no room for differentials in the E(4)local Adams
spectral sequence for the SmithToda complex V(3), so the cohomology of
the height 4 Morava stabilizer group computed in this paper is also
(after a regrading) the homotopy groups of the K(4)local SmithToda
V(3). I'm told that Mahowald conjectured that V(3) is the last
SmithToda complex to exist, i.e., that V(n) fails to exist for n > 3
and for all primes. If that conjecture is true, then the computation in
this paper finishes the problem of computing the K(n)local homotopy
groups of V(n1) for all n.
Other papers in pure mathematics.

Products in spin^ccobordism, joint with Hassan Abdallah, 2024.
An old problem of Stong from the 1960s asks to calculate
the spin^ccobordism ring, in particular, the multiplication on the
2torsion elements. (All torsion in the spin^ccobordism ring is known
to be 2torsion.) An inductive formula for the additive structure of the
spin^ccobordism ring is known from the work of Anderson, Brown, and
Peterson, but the multiplication, especially on the torsion elements,
has remained mysterious. In this paper, Hassan Abdallah and I calculate
the mod 2 spin^ccobordism ring up to inseparable isogeny, and we
calculate the spin^ccobordism ring "on the nose" in degrees up through
33. Even if you don't care about the product on spin^ccobordism, our
ringtheoretic descriptions wind up giving more digestible descriptions
of the underlying graded abelian group of the spin^ccobordism ring than
the rather mysterious Poincare series that you get from
AndersonBrownPeterson. Along the way, we use our methods to construct a
spin manifold with prescribed Pontryagin numbers which Milnor asked
about in the 1960s.

Derived functors of product and limit in the category of comodules over the dual Steenrod algebra, 2023.
In the 2000s, Sadofsky constructed a spectral sequence which
converges to the mod p homology groups of a homotopy limit of a
sequence of spectra. The input for this spectral sequence is the derived
functors of sequential limit in the category of graded comodules over
the dual Steenrod algebra. Since then, there has not been an
identification of those derived functors in more familiar or computable
terms. Consequently there have been no calculations using Sadofsky's
spectral sequence except in cases where these derived functors are
trivial in positive cohomological degrees.
In this paper I prove that the input for the Sadofsky
spectral sequence is simply the graded local cohomology of the Steenrod
algebra, taken with appropriate (quite computable) coefficients. This
turns out to require both some formal results, like some general results
on torsion theories and local cohomology of noncommutative
nonNoetherian rings, and some decidedly nonformal results, like a 1985
theorem of Steve Mitchell on some very specific duality properties of
the Steenrod algebra not shared by most finitetype Hopf algebras. Along
the way there are a few results of independent interest, such as an
identification of the category of graded A_*comodules with the full
subcategory of graded Amodules which are torsion in an appropriate
sense.

The Steenrod algebra is selfinjective, and the Steenrod algebra is not selfinjective, 2023.
It's wellknown that the Steenrod algebra is selfinjective, or to be careful, it's injective when regarded as a graded module over itself. In this paper I make the observation that the Steenrod algebra is not selfinjective as an ungraded
module over itself. This leads into a study of when the coproduct of
gradedinjective modules, over a general graded ring, remains
gradedinjective. I give a complete solution to that question by proving
a graded generalization of Carl Faith's characterization of
Sigmainjective modules. Specializing again to the Steenrod algebra, I
use that graded generalization of Faith's theorem to prove that the
covariant embedding of graded A_*comodules into graded Amodules
preserves injectivity for boundedabove objects, but not injectivity in
general.

An algebraic approach to asymptotics of the number of unlabelled bicolored graphs, 2024.
This is a paper in combinatorics, which establishes an
asymptotic count which I needed for another paper in another subject
(currently in progress). I think the methods are interesting in their
own right: the paper develops some basic theory of Dirichlet characters
on permutation groups, as well as a curious bilinear form on group
algebras of permutation groups, and uses these tools to help give a
wellstructured approach to the desired asymptotic count.

Kunneth formulas for Cotor, 2022.
In this paper I investigate the question of how to compute
the cotensor product, and more generally the derived cotensor (i.e.,
Cotor) groups, of a tensor product of comodules. I work out the
conditions under which there is a Kunneth formula for Cotor. It turns
out that there is a simple Kunneth theorem for Cotor groups if and only
if an appropriate coefficient comodule has trivial coaction. This result
is an application of a spectral sequence, constructed in this paper,
for computing Cotor of a tensor product of comodules. For certain
families of nontrivial comodules which are especially topologically
natural, I work out necessary and sufficient conditions for the
existence of a Kunneth formula for the 0th Cotor group, i.e., the
cotensor product. This has some topological applications to the E_2term
of the Adams spectral sequence of a smash product of spectra, and the
Hurewicz image of a smash product of spectra, and I explain a bit about
those applications in the paper.

Commuting unbounded homotopy limits with Morava Ktheory,
joint with Gabe AngeliniKnoll, originally 2020, although the linked
version is a revised version from 2023 with a greatly streamlined
argument.
An old theorem of Frank Adams establishes that homology
commutes with homotopy limits of sequences of spectra, as long as there
is a uniform lower bound on the connectivity of the spectra. This
doesn't give you any traction on homotopy limits of sequences of spectra
without that lower bound, however. In general, without any kind of
hypotheses, it's simply not true that a generalized homology theory will
commute with an unbounded homotopy limit.
In this paper, Gabe and I work out conditions under which
Morava Ktheory does commute with sequential homotopy limits, including
sequences of spectra that are not uniformly bounded below. As one
application, we prove the K(n)local triviality (for sufficiently large
n) of the algebraic Ktheory of algebras over truncated BrownPeterson
spectra, building on work of Bruner and Rognes and extending a classical
theorem of Mitchell on K(n)local triviality of the algebraic Ktheory
spectrum of the integers for large enough n.

A MilnorMoore theorem for bialgebras, joint with Joey BeauvaisFeisthauer and Yatin Patel, 2022.
Over fields of characteristic zero, in this paper Joey and
Yatin and I construct equivalences between certain categories of
bialgebras which are generated by grouplikes and generalized primitives,
and certain categories of structured Lie algebras. The relevant
families of bialgebras include many which are not connected, and which
fail to admit antipodes. Joey and Yatin and I have some applications in
mind for the MilnorMoore theorem from this paper, particularly to
equivariant formal group laws. We are presently working on a sequel
paper about those applications.

Topological Hecke eigenforms, joint with Luca Candelori, 2021 (although the linked version is a slightly improved revised version from 2024).
In this paper, Luca and I study the eigenforms of the
action of Andy Baker's Hecke operators on the holomorphic elliptic
homology of various topological spaces. We prove a multiplicity one
theorem (i.e., onedimensionality of the space of the topological Hecke
eigenforms for any given eigencharacter) for some classes of topological
spaces, and we give examples of finite CWcomplexes for which
multiplicity one fails. Classical multiplicity one theorems are really
fundamental and useful for the number theorists who use classical
modular forms. As far as I know, multiplicity one phenomena for
topological modular forms haven't been studied outside of this paper.
We also develop some abstract "derived eigentheory" whose
motivating examples arise from the failure of classical Hecke operators
to commute with multiplication by various Eisenstein series. Part of
this "derived eigentheory" is an identification of certain derived Hecke
eigenforms as the obstructions to extending topological Hecke
eigenforms from the top cell of a CWcomplex to the rest of the
CWcomplex. Using these obstruction classes together with our
multiplicity one theorem, we calculate the topological Hecke eigenforms
explicitly, in terms of pairs of classical modular forms, on all 2cell
CW complexes obtained by coning off an element in \pi_n(S^m) which
stably has AdamsNovikov filtration 1.

The topological Petersson product, joint with Luca Candelori, 2021.
In this note, Luca and I prove that there's no extension of
the Petersson product on classical cusp forms to a reasonable product
on topological cusp forms. Part of what "reasonable" entails here is
nondegeneracy, since that's an essential property of the classical
Petersson product: it's the slick way to show that classical cusp forms
have a basis consisting of Hecke eigenforms, and you'd like to be able
to do the same for topological cusp forms. The trouble is that we show
that the obvious way that the Petersson product ought to act on rational
tcf results in a degenerate inner product for all but the simplest
spaces. So even rationally there's no hope for a topological Petersson
product which could be used to prove diagonalizability of the action of
Baker's Hecke operators on tmf or tcf of a large class of spaces.

A Maytype spectral sequence for higher topological Hochschild homology, joint with Gabe AngeliniKnoll, 2016.
Given a filtration of a commutative monoid A in a symmetric
monoidal stable model category C, we construct a spectral sequence
analogous to the May spectral sequence whose input is the higher order
topological Hochschild homology of the associated graded commutative
monoid of A, and whose output is the higher order topological Hochschild
homology of A. We then construct examples of such filtrations and
derive some consequences: for example, given a connective commutative
graded ring R, we get an upper bound on the size of the THHgroups of
E_inftyring spectra A such that pi_*(A) = R. Gabe uses this spectral
sequence in his computation of the topological Hochschild homology of
the algebraic Ktheory spectrum of certain finite fields, which is input
for his further computations in the iterated algebraic Ktheory of
finite fields. You can find Gabe's computation of the V(1)homotopy of
THH(K(F_q)) using our THHMay spectral sequence, for a certain family of
prime powers q, in the paper On topological Hochschild homology of the K(1)local sphere on Gabe's website, here.

Graded comodule categories with enough projectives, written 2016.
It is wellknown that the category of comodules over a flat
Hopf algebroid is abelian but typically fails to have enough
projectives, and more generally, the category of graded comodules over a
graded flat Hopf algebroid is abelian but typically fails to have
enough projectives. In this short paper we prove that the category of
connective graded comodules over a connective, graded, flat, finitetype
Hopf algebroid has enough projectives. In particular, the Hopf
algebroids of stable cooperations in complex bordism, BrownPeterson
homology, and classical mod p homology all have the property that their
categories of connective graded comodules have enough projectives.

The Hochschild homology and cohomology of A(1),
2015, although the version here is a revision from 2023 incorporating a
few improvements, like a duality argument that yields the Hochschild
cohomology as well as homology.
This paper solves an old "folk" problem, the computation of
the Hochschild homology of A(1), the subalgebra of the 2primary
Steenrod algebra generated by the first two Steenrod squares, Sq^1,Sq^2.
The main tool in the computation is a number of Maytype spectral
sequences.

The Bousfield localizations and colocalizations of the discrete model structure,
2016. (The earliest version of this paper was another one of the first
things I wrote, but in 2016 I went back and rewrote the paper from
scratch.)
This short paper works out all the Bousfield localizations
and colocalizations of the discrete model structure on any reasonable
category, including the homotopy category and Waldhausen Ktheory of
each of those (co)localizations. As an application, it shows that every
replete reflective subcategory is the subcategory of fibrant objects of a
model structure. (The results in this particular paper aren't difficult
to prove: most of the work is done by a theorem of Cassidy, Hebert, and
Kelly. But these results apparently aren't already in the literature,
and several other homotopy theorists have told me they found the results
novel, so I went ahead and submitted the paper.)

Totalization of simplicial homotopy types, joint with Crichton Ogle, 2013.
It is fairly wellknown that, given a simplicial object in
the homotopy category of a stable model category (i.e., a "simplicial
homotopy type"), there are obstructions to building the geometric
realization of that simplicial homotopy type; these obstructions are
given by Toda brackets; and these obstructions vanish, of course, if the
simplicial object lifts to a simplicial object in the stable model
category itself.
Now given two simplicial homotopy types which are
geometrically realizable, and a map of simplicial homotopy types between
them, can that map be geometrically realized? In this paper we build a
sequence of "secondary Toda brackets" which are the obstructions to
geometric realization of a map of simplicial homotopy types. We also
give an example from cyclic homology, due to Ogle, where these
obstructions do not vanish.

Relative homological algebra, Waldhausen Ktheory, and quasiFrobenius conditions, 2013.
An "allowable class" on an abelian category is the
structure required to do relative homological algebra in that category.
(If you have ever worked with comodules over Hopf algebroids in order to
compute generalized Adams spectral sequences, then this is a familiar
story, the generalized Adams E_2term is a relative Ext.) This paper
works out when an allowable class on an abelian category also defines a
Waldhausen structure with cylinder functor, so that one can consider and
compute algebraic Kgroups of that category which split all the short
exact sequences in that allowable class.
As an application, this paper proves the following: let k
be a finite field with n elements, and let C be the category of finitely
generated k[x]/x^nmodules, with the Waldhausen structure in which the
cofibrations are the monomorphisms and the weak equivalences are the
stable equivalences. Let K(C) denote the Waldhausen Ktheory spectrum of
C. Then K(C) is a complexoriented ring spectrum. This is a partial
answer to a question asked by J. Morava about which Ktheory spectra
admit complex orientations.

Homotopy colimits in stable representation theory, 2013.
In representation theory of algebras (and, for a stable
homotopy theorist, the motivating examples are often the Steenrod
algebra or its subalgebras, or the group ring of the Morava stabilizer
group or its subalgebras) one encounters the following construction:
given an abelian category C, two maps are homotopic if their difference
factors through a projective object, and a map is a stable equivalence
if it admits an inverse up to homotopy. If C is quasiFrobenius, then
there exists a model structure on C whose weak equivalences are the
stable equivalences, and one can make use of the usual theory of
homotopy colimits in a model category. If C is not quasiFrobenius, then
it is not known that C admits a model structure in which the weak
equivalences are the stable equivalences, and so it is not clear whether
C admits homotopyinvariant colimits. This paper proves that the answer
is no: if C has enough projectives and at least one object of
finite, positive projective dimension, then C does not admit
welldefined homotopy cofibers. Consequently C does not admit a model
structure in which the weak equivalences are the stable equivalences.
On the other hand, when C has enough projectives, when
every projective object in C is injective, and when every object can be
embedded appropriately into a projective object, then in this paper it
is proven that C does admit homotopyinvariant colimits. This
includes cases where it is not known that C admits a model category
structure whose weak equivalences are stable equivalences. The results
in the paper are also somewhat more general than this, allowing the use
of relative projectives (in the sense of relative homological algebra)
in places of projectives, throughout.

A recognition principle for the existence of descent data, 2014.
Let R and S be rings. Given a faithfully flat ring map from
R to S and an Rmodule M, wellknown descent theory describes (in terms
of a cohomology group, for example) the set of isomorphism classes of
S/Rdescent data on M basechanged to S, i.e., the set of isomorphism
classes of Rmodules whose basechange to S is isomorphic to that of M.
But, given an Smodule N, there do not seem to be any tools for
determining whether N admits a S/Rdescent datum at all. In other words:
given an Smodule N, how can we tell whether N is isomorphic to some
Rmodule basechanged to S?
This paper develops a tool for answering questions of this
kind. The main theorem is phrased in a high level of abstraction, in
terms of extensions of comonads, but the concrete special case I like
best is the following: given an augmented commutative algebra A over a
field k and a ring map from A to a commutative kalgebra B, we get
reasonable sufficient conditions on the map for the following condition
to hold: a Bmodule M admits an B/Adescent datum if and only if the
basechange of M to B \tensor_A k is a free (B \tensor_A k)module.This
version of the paper also includes an example computation at the end
which did not appear in the journal version (the referee thought it was
redundant; I see the point, but I still prefer to give example
computations).

How many adjunctions give rise to the same monad?, 2014.
Given an adjoint pair of functors F,G, the composite GF
naturally gets the structure of a monad. The same monad may arise from
many such adjoint pairs of functors, however. This paper works out some
methods for computing the collection of adjunctions which give rise a
given monad. This includes a Becklike criterion for uniqueness of such
adjunctions, and also some explicit example computations.

Obstructions to compatible splittings, 2014.
If one has a map of split short exact sequences in a
category of modules, or more generally, in any abelian category, then it
is not always the case that the short exact sequences split compatibly. In this paper we define and prove basic properties of a group of obstructions to the existence of compatible splittings.

Failure of flat descent of model structures on module categories, 2014.
This one isn't published or submitted for publication; it's
just a short note with one small interesting result: if you define a
"presheaf of model structures on module categories" which sends a scheme
(or stack) X to the collection of model structures on the category of
quasicoherent Xmodules, then this presheaf fails to satisfy fppf
descent.
Papers on data and fMRI.
For several years I have worked on applications of methods from higher
mathematics to fMRI data, in collaboration with the Diwadkar lab in
WSU's medical school, as well as with my students and collaborators
within the math department. This is a long way from the pure mathematics
projects I largely focus on, but I suspect it's probably good for a
researcher to think about a mix of pure questions and applied questions.
In the beginning our group's focus was narrowly on the
application of topological tools (particularly persistent homology) to
fMRI, but more recently we have taken a broader view, using simplicial
methods from topology as well as techniques of statistical inference to
develop some new tools for higherorder network discovery and analysis.
The papers we have finished so far are below.

From
mathematics to medicine: A practical primer on topological data
analysis (TDA) and the development of related analytic tools for the
functional discovery of latent structure in fMRI data, 2021, joint with Regalski, Abdallah, Suryadevara, Catanzaro, and Diwadkar.
This paper is intended to explain the basic ideas of
persistent homology to neuroimaging researchers, and to present a
practical workflow for actually calculating persistent homology of fMRI
data. We give links to free software packages written by my students
Adam Regalski and Hassan Abdallah, so neuroimaging researchers can
easily use the tools on their own data.

Statistical inference for persistent homology applied to simulated fMRI time series data, 2022, joint with Abdallah, Regalski, Kang, Berishaj, Nnadi, Chowdury, and Diwadkar.
This paper presents a Monte Carlo test for persistent
homology applied individually to each time index in time series data.
The idea is that you'd like to compare one set of persistence diagrams
to another set of persistence diagrams, and use a statistical test to
infer that the persistence diagrams in the first set are significantly
different than those in the other set.
Notes.
Luca Candelori and I organized a weekly seminar on modular forms, here
in the mathematics department at WSU, during the 20182019 academic
year. One major goal of the seminar was to cover enough background from
number theory and topology for us to discuss two open, probably rather
difficult, questions: 1. whether one can give Ktheoretic or
homotopytheoretic descriptions of special values of Lfunctions of
eigenvalue 1/4 Maass cusp forms, and 2. whether there exist spectral
enrichments of geometric models for Maass forms (of some fixed level and
eigenvalue), i.e., whether there exist "topological Maass forms,"
especially in the harmonic case and the eigenvalue 1/4 case.
Below are some of our notes from the seminar. We made some
effort to include the numbertheoretic content that readers with a
background in homotopy theory are less likely to already know, and the
homotopytheoretic content that readers with a background in number
theory are less likely to already know.