Research
The papers and preprints below are organised by topic. I have comments in the abstracts of some of them.
Homotopy theory
- Enriched coalgebras are sometimes comonadic Submitted (ArXiv) Abstract
This is the first submitted part of my thesis chapter "A higher Hochschild-Konstant-Rosenberg Theorem and the Deligne conjecture": We introduce an enriched notion of coalgebras over V-operad P in a
symmetric monoidal V-category C. When C is semicartesian, we construct an
endofunctor on C associated to P and give conditions under which it is a
comonad with co-Eilenberg--Moore category equivalent to the category of
enriched P-algebras in V. In many cases, this permits easy computation of
V-categories of coalgebras. We give several simple examples and show that our
theorem generalises one direction of a well-known theorem of Fox..
- A p-adic de Rham complex (ArXiv) NOT SUBMITTED (see note) Abstract
[I'm very, very sorry about this, but there are serious issues with the proof of Lemma 3.11 (precisely: there is an extra cocycle in degree 1 part of the 0-cocycles in the base case of the induction of the form x_0^{[2]}-x_0 - this doesn't die because x_0x_1 is not divisible by 2 and so creates a 2-torsion in the homotopy groups of the 0-cocycles) in this manuscript. It took me six months to locate the precise source of this error: I think the results are still correct "up to minor change of definition" (morally you just have to kill torsion components as above by adding a "closure under rewriting" rule allowing you to write x_0^{[2]}-x_0 -> 1/2x_0x_1. I already have a closure condition in the paper, but this is very vaguely phrased and I need to be much more precise to ensure errors of this kind are avoided.). I am currently preparing a much more detailed and longer version of the manuscript. Unfortunately I have to deal with some other projects first, but I hope to have a clean version by the end of 2026.] This is the second in the sequence of three articles exploring the relationship between
commutative algebras and E∞-algebras in characteristic p and mixed characteristic. Given a
topological space X, we construct, in a manner analogous to Sullivan's APL-functor, a strictly
commutative algebra over the p-adic numbers which we call the de Rham forms on X. We show this complex
computes the singular cohomology ring of X. We prove that it is quasi-isomorphic as an
E∞-algebra to the Berthelot-Ogus-Deligne décalage of the singular cochains complex with
respect to the p-adic filtration. We show that one
can extract concrete invariants from our model, including Massey products which live in the
torsion part of the cohomology. We show that if X is formal then, except at possibly finitely
many primes, the p-adic de Rham forms on X are also formal.
- An obstruction theory for strictly commutative algebras in positive characteristic (journal, ArXiv) Algebraic and Geometric Topology (2026)
Abstract Slides
This is chapter 4 in my thesis, which was part of collection looking into the relationship between commutative algebras and E∞ algebras in characteristic p and mixed characteristic. I thought this was a cool topic to study because it seemed to be simplest context in which one needs new ideas from outside the classical operadic calculus to attack. It turned out that some old tools could be tweaked a bit; you could define "higher Massey products for the Frobenius maps" which I called cotriple products as they represent differentials in the cotriple resolution. In a recent Acta paper, Campos, Petersen, Robert-Nicoud and Wierstra had asked a question about commutative and associative quasi-isomorphisms and we were able to build algebras around these invariants in order to answer the question in positive characteristic (negatively of course!). Finally, the main theorem stated that E∞ algebras are weakly equivalent to commutative algebras if and only if the higher Steenrod operations vanish *coherently*. The necessity of this coherence condition was my main disappointment with the paper - starting the project I had initially hoped that it was more pleasant than formality. On the plus side, I guess this leaves open the possibility of constructing faithful obstructions to future work.
- Higher order Massey products for algebras over algebraic operads (with José M. Moreno-Fernández) (ArXiv) Submitted
Abstract
We introduce higher-order Massey products for algebras over algebraic operads. This extends the work of Fernando Muro on secondary ones. We study their basic properties and behavior with respect to morphisms of algebras and operads and give some connections to formality. We prove that these higher-order operations represent the differentials in a naturally associated operadic Eilenberg--Moore spectral sequence. We also study the interplay between particular choices of higher-order Massey products and quasi-isomorphic P∞-structures on the homology of a P-algebra. We focus on Koszul operads over a characteristic zero field and explain how our results generalize to the non-Koszul case.
- A recognition principle for iterated suspensions as coalgebras over the little cubes operad (with José M. Moreno-Fernández and Felix Wierstra)
(ArXiv) Submitted Abstract
Our main result is a recognition principle for iterated suspensions as coalgebras over the little disks operads. Given a topological operad, we construct a comonad in pointed topological spaces endowed with the wedge product. We then prove an approximation theorem that shows that the comonad associated to the little n-cubes operad is weakly equivalent to the comonad ΣⁿΩⁿ arising from the suspension-loop space adjunction. Finally, our recognition theorem states that every little n-cubes coalgebra is homotopy equivalent to an n-fold suspension. These results are the Eckmann--Hilton dual of May's foundational results on iterated loop spaces.
- Simplicial coendomorphism operads and coalgebras (journal), Bulletin of the Australian Mathematical Society Abstract
In recent work of Moreno-Fernandez, Wierstra and the author, a coendomorphism operad in the category of pointed topological spaces endowed with the wedge sum was introduced. In this paper, using Kan's $\exi$-functor, we construct an analogue completely internal to the category of simplicial sets with the goal of defining simplicial coalgebras. As an application, we show that simplicial $n$-fold suspensions are coalgebras up to coherent homotopy over the Barratt--Eccles $E_n$-operad.
- Three Schur functors related to pre-Lie algebras (with Vladimir Dotsenko) (journal, ArXiv ) Mathematical Proceedings of Cambridge Philosophical Society (2024)
Abstract
We give explicit combinatorial descriptions of three Schur functors arising in the theory of pre-Lie algebras. The first of them leads to a functorial description of the underlying vector space of the universal enveloping pre-Lie algebra of a given Lie algebra, strengthening the PBW theorem of Segal. The two other Schur functors provide functorial descriptions of the underlying vector spaces of the universal multiplicative enveloping algebra and of the module of Kähler differentials of a given pre-Lie algebra. An important consequence of such descriptions is an interpretation of the cohomology of a pre-Lie algebra with coefficients in a module as a derived functor for the category of modules over the universal multiplicative enveloping algebra.
Theoretical Computer Science
- Central limits via dilated categories (with Henning Basold, Chase Ford and Hao Wang) (ArXiv) Submitted Abstract
Our main result is a categorical version of the central limit theorem. Additionally, we introduce a new category to act as a base of enrichment and prove a categorical version of the Banach fixed point theorem.
Number Theory
- On the divisibility of sums of Fibonacci numbers (journal, ArXiv, errata), Integers: Electronic Journal of Combinatorial Number Theory (2025) Abstract
We show that for infinitely many odd integers $n$, the sum of the first $n$ Fibonacci numbers is divisible by $n$. This resolves a conjecture of Fatehizadeh and Yaqubi. Comments: As of February 2026, this is the fastest paper I have ever written; it took only about a weekend after becoming obsessed with the problem (after randomly happening upon while googling something else). It is also the only paper that I have ever used a computer for - after guessing recursion was key, I used Python to find some examples and eventually to spot the pattern the paper is based on. Errata: The statement of Theorem 1 (on the behaviour of $S_p$ for primes $p > 5$) is off by a constant factor due to an arithmetic error; see the errata for the corrected statement. The main result of the paper is unaffected.
Expository
- On associative and commutative differential graded algebras in positive characteristic (journal, local), Mathematical Intelligencer (2026) Abstract
This is an expository paper, explaining one of the constructions in "An obstruction theory for strictly commutative algebras in positive characteristic" to the general reader.
In preparation
- A Banach fixed point theorem for lattice valued metrics (CS) with Chase Ford and Henning Basold
- Coalgebras in abelian groups (CS) with Chase Ford and Henning Basold
- Determinant Massey products (Maths) with Jose Moreno-Fernandez and Fernando Muro
- A higher Hochschild-Konstant-Rosenberg Theorem and the Deligne conjecture (Maths)
Ongoing projects
- A categorical approach to optimization (with Henning Basold, Chase Ford and Hao Wang) Abstract
This project aims to study optimization processes through the lens of enriched category theory. First, we developed machinery for understanding the Central Limit Theorem and analytic convergence in an algebraic and categorical context. Second, we will apply this to model Stochastic Differential Equations within the category of diffeological spaces. Finally, we hope to conclude the project by integrating homotopy theory in order to study homotopical optimization processes.
