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 up-to-date 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.

1. "Complex multiplication and homotopy groups of spheres" project, with subtopics:
   • Papers on complex-analytic L-functions (e.g. automorphic L-functions) associated to finite CW-complexes
   • Papers on Iwasawa theory and p-adic L-functions associated to finite CW-complexes
   • Papers on moduli of formal modules and their applications
   • Papers which carry out calculations in stable homotopy theory using or used by papers from the above sections


2. Other papers in pure mathematics


3. Papers on data analysis and fMRI


4. Notes


5. Software

"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 Bousfield-localized finite spectra in terms of special values of L-functions and zeta-functions that arise in number theory. The broad motivation for such a project is that it is generally extremely difficult to even describe the v_n-periodic 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 well-studied elsewhere in mathematics--in particular, special values of some L-function or zeta-function.

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_K-modules have orders describable in terms of the Dedekind zeta-function of K (e.g. see the "Structure and cohomology..." paper, below). Formal modules also expedite some "height-shifting" tricks, since if K is a degree d extension of the p-adic rationals, then the automorphism group of an O_K-height n/d formal O_K-module 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 A-modules 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 complex-analytic L-functions (e.g. automorphic L-functions) associated to finite CW-complexes,
• papers on Iwasawa theory and p-adic L-functions associated to finite CW-complexes,
• papers on moduli of formal modules and their applications, e.g. via "height-shifting,"
• and papers which carry out calculations in stable homotopy theory which are used to prove results in any of the above topics.



• Papers on complex-analytic L-functions associated to finite CW-complexes:

  • Denominators of special values of zeta-functions count KU-local 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 KU-local homotopy groups of the mod p Moore spectrum are equal to denominators of special values of certain quotients of Dedekind zeta-functions of totally real number fields. I use this and Colmez's p-adic 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 KU-local stable homotopy groups.
  
  
  • KU-local zeta-functions of finite CW-complexes, 2023.
    
    This paper extends the main result "Denominators of special values...mod p Moore spectra" by defining a KU-local zeta-function for any finite CW-complex 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 L-functions of characters, and showing that its special values at negative integers have denominators equal to the orders of the KU-local stable homotopy groups of X, up to powers of certain primes.
  
  
  • ell-adic topological Jacquet-Langlands 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 Lubin-Tate spaces in both computational homotopy theory (since the cohomology of the Morava stabilizer group scheme with coefficients in the global sections on Lubin-Tate 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 Jacquet-Langlands correspondences. Carayol essentially conjectured that the ell-adic vanishing cycle cohomology of the Lubin-Tate tower would realize the Langlands and Jacquet-Langlands correspondences in a certain precise way. This conjecture was later proven by Henniart and by Harris-Taylor. An old sticking point, for making any fruitful connection to homotopy theory: Schwanzl-Vogt-Waldhausen showed that the realization of the base of the Lubin-Tate tower by an E_\infty-ring spectrum can't possibly extend to a realization of the higher stages in the Lubin-Tate tower.
    
    In the appendix of this paper, Matthias Strauch and I describe our workaround for that old problem: we embed the Lubin-Tate tower into a larger tower of deformation spaces, which we call "degenerating Lubin-Tate spaces." We use Hopkins-Kuhn-Ravenel to construct a spectral realization of the degenerating Lubin-Tate spaces, and we show that the relevant vanishing cycle cohomology of the classical Lubin-Tate spaces splits off as a summand in the cohomology of the degenerating Lubin-Tate 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 Lubin-Tate tower to construct "ell-adic Jacquet-Langlands cohomology". This is a generalized cohomology theory which realizes the ell-adic Jacquet-Langlands and ell-adic Langlands correspondences for Q_p in its rational stable homotopy groups. I use it to build an "ell-adic topological Jacquet-Langlands (TJL) dual" of any spectrum with respect to any one-dimensional 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 ell-adic Jacquet-Langlands duals of representations of Aut(G) occurring in the Morava E-theory E(G)_*(X) of X. Finally, if G has height 1, I show that the product of the automorphic L-factors associated to the GL_1-representations occuring in the TJL dual of X are precisely the "provisional KU-local zeta function of X" as defined in the "KU-local zeta-functions of finite CW-complexes" paper, above. Consequently this Euler product analytically continues to a meromorphic function on the complex plane, and its special values in the left half-plane recover the orders of the KU-local 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 p-adic L-functions associated to finite CW-complexes:

  • 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 p-adically complete complex K-theory of finite CW-complexes, for an odd prime p. The resulting Iwasawa invariants are topologically meaningful: for example, we show that the Iwasawa lambda-invariant measures the asymptotic growth rate of the K(1)-local homotopy groups of the CW-complex. The characteristic polynomials yield algebraic p-adic L-functions of finite CW-complexes, and we prove a "weak form of the Iwasawa main conjecture" in this setting, which asserts that the special values of this algebraic p-adic L-function at negative integers have p-adic valuations which agree, up to sign, with the p-adic valuations of the orders of the K(1)-local homotopy groups of the CW-complex. In many ways this paper does for p-adic L-functions what the "KU-local zeta-functions of finite CW-complexes" paper does for complex-analytic zeta-functions.
  
  

• Papers on moduli of formal modules and their applications:

  • The action of the Morava stabilizer group on the coefficients of Morava E-theory at height 2, 2025.
    
    In this paper, I use ideas of Devinatz-Hopkins and some combinatorial tricks to calculate an explicit closed formula for the action of the full height 2 Morava stabilizer group on the coefficient ring of the height 2 Morava E-theory. The essential idea here, which lets this paper get further than various earlier papers (surveyed in section 1) which calculate approximations to the action, is to use labelled trees to organize the summands in the calculation. A Sage script which I wrote for making related calculations is available further down on this page.
  
  
  • 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 A-modules and about the moduli stack M_{fmA} of formal A-modules. I use these structural results to aid in explicit calculations of flat cohomology groups of M_{fmA}^{2-buds}, the moduli stack of formal A-module 2-buds. For example, a generator of the group H^1_{fl}(M_{fm Z}; omega), which also generates (via the Adams-Novikov spectral sequence) the first stable homotopy group of spheres, also yields a generator of the A-module H^1_{fl}(M_{fmA}^{2-buds}; omega) for any torsion-free Noetherian commutative ring A. I show that the order of the A-modules H^1_{fl}(M_{fmA}^{2-buds}; omega) and H^2_{fl}(M_{fmA}^{2-buds}; omega tensor omega) are each equal to 2^{N_1}, where N_1 is the leading coefficient in the 2-local zeta-function of Spec A.
    
    In this paper I also prove that the cohomology of M_{fmA}^{2-buds} is closely connected to the delta-invariant and syzygetic ideals studied in commutative algebra: H^0_{fl}(M_{fmA}^{2-buds}; omega tensor omega) is the delta-invariant 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}^{2-buds}; 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 A-modules, 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 A-module. 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 A-modules, namely, any formal A-module over R/I lifts to a formal A-module over R, and any formal A-module n-bud extends to a formal A-module (n+1)-bud.
  
  
  • Moduli of formal A-modules 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 p-adic 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/Lubin-Tate 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 Smith-Toda complex V(1) to the (automatically K(2)-local) homotopy groups of the homotopy fixed-points 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 Milnor-Moore 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 Chevalley-Eilenberg 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 Devinatz-Hopkins to take the homotopy fixed-points of the action of this closed subgroup on a Lubin-Tate/Morava E-theory 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 Smith-Toda 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 Lubin-Tate 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 A-modules 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 p-adic 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 p-adic integers, S^A exists: it is simply the p-complete sphere spectrum.Some later work was done on this problem: A. Pearlman computed the Morava K-theories of the spectra S^A, assuming they exist, and T. Lawson showed that, if S^A exists, then it cannot be a A_p-ring spectrum, that is, a spectrum with a ring structure that is associative up to pth-order homotopies. Lawson also showed that S^A does not exist if A is the ring of integers in the 2-adic 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 p-adic integers. The natural global and p-typical 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 Brown-Peterson homology, which implies that the only spectra X such that BP_*(X) is a V^A-module are extensions of rational spectra by dissonant spectra. Here V^A is the classifying ring of A-typical formal A-modules.
    
    While formal A-modules 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 Smith-Toda 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 A-modules, or equivalently, the orders of the flat cohomology groups H^1 of the moduli stack of formal A-modules, 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 L-functions 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 A-modules is isomorphic (as a graded abelian group) to the flat cohomology group of the moduli stack of formal B-modules if and only if the Dedekind zeta-function of K is equal to the Dedekind zeta-function 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 Lubin-Tate spaces, and more "globally," Shimura varieties, are often used to study the zeta-functions associated to those representations. The moduli stack of formal A-modules is an alternative "globalization" to a Shimura variety: like a Shimura variety, the formal completion at each point is a deformation space of a Barsotti-Tate module (e.g. a Lubin-Tate space), and in this paper we show that zeta-function data is also visible in the cohomology of the moduli stack of formal A-modules.
  
  

• 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 recently-graduated 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_2-page for the E(n)-local Adams spectral sequence converging to the E(n)-local homotopy groups of the Smith-Toda complex V(n-1). 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^n-1) stems in the E_2-page 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 Chevalley-Eilenberg DGA of Ravenel's Lie algebra model L(n,n) for the cohomology of the Morava stabilizer group. The smooth fibers are each Chevalley-Eilenberg DGAs of a reductive Lie algebra isomorphic to gl_n(F_p). We show there is an essentially-unique C_n-equivariant multiplicative differential-graded connection on the resulting bundle of DGAs over A^1 - {0}. From parallel transport in this connection, you get a Picard-Lefschetz (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 T-fixed 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 Chevalley-Eilenberg 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 finite-dimensional solvable Lie algebra. That means the cohomology is computable algorithmically, since there's a "brute force" algorithm for calculating cohomology of finite-dimensional Lie algebras (just calculate the cohomology of the Chevalley-Eilenberg complex).
    
    One consequence is the existence, for a prime p>>n and any E(n-1)-acyclic finite CW-complex 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 p-primary 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 write-up 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 p-height 4 formal group law over the field with p elements, at primes p > 5. This uses some new "height-shifting" 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 A-height n formal A-module, where A is the ring of integers in a quadratic extension of the p-adic rationals; and then we construct a sequence of spectral sequences whose input is the cohomology of an A-height n formal A-module, 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 Smith-Toda 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 Smith-Toda V(3). I'm told that Mahowald conjectured that V(3) is the last Smith-Toda 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(n-1) for all n.
  
  
  

  Other papers in pure mathematics.

  • Products in spin^c-cobordism, joint with Hassan Abdallah, 2024.
    
    An old problem of Stong from the 1960s asks to calculate the spin^c-cobordism ring, in particular, the multiplication on the 2-torsion elements. (All torsion in the spin^c-cobordism ring is known to be 2-torsion.) An inductive formula for the additive structure of the spin^c-cobordism 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^c-cobordism ring up to inseparable isogeny, and we calculate the spin^c-cobordism ring "on the nose" in degrees up through 33. Even if you don't care about the product on spin^c-cobordism, our ring-theoretic descriptions wind up giving more digestible descriptions of the underlying graded abelian group of the spin^c-cobordism ring than the rather mysterious Poincare series that you get from Anderson-Brown-Peterson. 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 non-Noetherian rings, and some decidedly non-formal results, like a 1985 theorem of Steve Mitchell on some very specific duality properties of the Steenrod algebra not shared by most finite-type 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 A-modules which are torsion in an appropriate sense.
  
  
  • The Steenrod algebra is self-injective, and the Steenrod algebra is not self-injective, 2023.
    
    It's well-known that the Steenrod algebra is self-injective, 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 self-injective as an ungraded module over itself. This leads into a study of when the coproduct of graded-injective modules, over a general graded ring, remains graded-injective. I give a complete solution to that question by proving a graded generalization of Carl Faith's characterization of Sigma-injective 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 A-modules preserves injectivity for bounded-above 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 well-structured 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_2-term 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 K-theory, joint with Gabe Angelini-Knoll, 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 K-theory 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 K-theory of algebras over truncated Brown-Peterson spectra, building on work of Bruner and Rognes and extending a classical theorem of Mitchell on K(n)-local triviality of the algebraic K-theory spectrum of the integers for large enough n.
  
  
  • A Milnor-Moore theorem for bialgebras, joint with Joey Beauvais-Feisthauer 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 Milnor-Moore 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., one-dimensionality of the space of the topological Hecke eigenforms for any given eigencharacter) for some classes of topological spaces, and we give examples of finite CW-complexes 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 CW-complex to the rest of the CW-complex. 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 2-cell CW complexes obtained by coning off an element in \pi_n(S^m) which stably has Adams-Novikov 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 May-type spectral sequence for higher topological Hochschild homology, joint with Gabe Angelini-Knoll, 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 THH-groups of E_infty-ring spectra A such that pi_*(A) = R. Gabe uses this spectral sequence in his computation of the topological Hochschild homology of the algebraic K-theory spectrum of certain finite fields, which is input for his further computations in the iterated algebraic K-theory of finite fields. You can find Gabe's computation of the V(1)-homotopy of THH(K(F_q)) using our THH-May 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 well-known 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, finite-type Hopf algebroid has enough projectives. In particular, the Hopf algebroids of stable co-operations in complex bordism, Brown-Peterson 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 2-primary Steenrod algebra generated by the first two Steenrod squares, Sq^1,Sq^2. The main tool in the computation is a number of May-type 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 K-theory 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 well-known 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 K-theory, and quasi-Frobenius 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_2-term 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 K-groups 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^n-modules, 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 K-theory spectrum of C. Then K(C) is a complex-oriented ring spectrum. This is a partial answer to a question asked by J. Morava about which K-theory 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 quasi-Frobenius, 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 quasi-Frobenius, 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 homotopy-invariant 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 well-defined 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 homotopy-invariant 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 R-module M, well-known descent theory describes (in terms of a cohomology group, for example) the set of isomorphism classes of S/R-descent data on M base-changed to S, i.e., the set of isomorphism classes of R-modules whose base-change to S is isomorphic to that of M. But, given an S-module N, there do not seem to be any tools for determining whether N admits a S/R-descent datum at all. In other words: given an S-module N, how can we tell whether N is isomorphic to some R-module base-changed 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 k-algebra B, we get reasonable sufficient conditions on the map for the following condition to hold: a B-module M admits an B/A-descent datum if and only if the base-change 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 Beck-like 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 X-modules, 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 higher-order 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 2018-2019 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 K-theoretic or homotopy-theoretic descriptions of special values of L-functions 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 number-theoretic content that readers with a background in homotopy theory are less likely to already know, and the homotopy-theoretic content that readers with a background in number theory are less likely to already know.
  • Lecture 1: Candelori on the Riemann zeta-function
  
  
  • Lecture 2: Salch on special values of zeta-functions
  
  
  • Lecture 3: Candelori on theta functions and modular forms
  
  
  • Lecture 4: Candelori on q-expansions and Waring's problem
  
  
  • Lecture 5: Candelori on geometric models for modular forms
  
  
  • Lecture 6: Salch on moduli and stacks
  
  
  • Lecture 7: Salch on formal groups, supersingularity, and L-functions
  
  

  Software.

  • Sage script for calculating the action of the height 2 Morava stabilizer group on the coefficients of Morava E-theory modulo various powers of u1, public version 1, 2025.