Oisín Flynn-Connolly

Research papers

Papers and preprints. Open an entry for details and notes.

• Homotopy theory Accepted

  Higher order Massey products for algebras over algebraic operads

  Oisín Flynn-Connolly and José M. Moreno-Fernández

  Mathematische Zeitschrift, to appear

  arXivSlides

  Abstract and notes

  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.

• Homotopy theory Published

  An obstruction theory for strictly commutative algebras in positive characteristic

  Oisín Flynn-Connolly

  Algebraic & Geometric Topology, 2026

  JournalarXivSeville slidesToulouse slidesLille slides

  Abstract and notes

  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 coherence condition leaves open the possibility of constructing faithful obstructions in future work.

• Homotopy theory Published

  Simplicial coendomorphism operads and coalgebras

  Oisín Flynn-Connolly

  Bulletin of the Australian Mathematical Society, 2026

  Journal

  Abstract and notes

  This was based on my master thesis. 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.

• Expository Published

  On associative and commutative differential graded algebras in positive characteristic

  Oisín Flynn-Connolly

  The Mathematical Intelligencer, 2026

  JournalPDFRelated slides

  Abstract and notes

  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.

• Number theory Published

  On the divisibility of sums of Fibonacci numbers

  Oisín Flynn-Connolly

  Integers: Electronic Journal of Combinatorial Number Theory, 2025

  JournalarXivErrataTalk video

  Abstract and notes

  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.

• Algebra and categorical algebra Published

  Three Schur functors related to pre-Lie algebras

  Vladimir Dotsenko and Oisín Flynn-Connolly

  Mathematical Proceedings of the Cambridge Philosophical Society, 2024

  JournalarXiv

  Abstract and notes

  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.

• Algebra and categorical algebra Submitted

  Enriched coalgebras are sometimes comonadic

  Oisín Flynn-Connolly

  Preprint, 2026

  arXiv

  Abstract and notes

  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.

• Algebra and categorical algebra Submitted

  On a conjecture of Dotsenko on weak nilpotence and the Engel identity

  Oisín Flynn-Connolly

  Preprint, 2026

  PDF

  Verification code (Macaulay2): 7-Engel verifier · 8-Engel verifier · 9-Engel certificate

  Abstract

  By means of AI-assisted computer algebra, we give examples of commutative but nonassociative algebras satisfying the identity t₄ = 0 such that neither the 7 nor 8-Engel identity holds. This resolves one conjecture of Dotsenko in the negative, and confirms a second. Conversely, we show that t₄ = 0 implies the 9-Engel identity. Together these show that 9 is the least integer m for which t₄ = 0 implies the m-Engel identity. Note: I had tried to prove some of the conjectures in the Nilpotence paper back in 2025 shortly after it came out. I had thought the conjecture was true and was very surprised when the desired relation turned out not to hold in the free algebra on two generators after I asked claude to construct it.

• Theoretical computer science Preprint

  Central limits via dilated categories

  Henning Basold, Oisín Flynn-Connolly, Chase Ford and Hao Wang

  Preprint, 2026

  arXivSlides

  Abstract and notes

  Note: I have a longer version (that proves a much better theorem and has more examples) that should be on the ArXiv soon and I'd be delighted to share it if you email me. 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.

• Homotopy theory Draft under revision

  A p-adic de Rham complex

  Oisín Flynn-Connolly

  arXiv manuscript, 2025

  arXiv

  The current arXiv manuscript has a serious issue in the proof of Lemma 3.11 and has not been submitted. I am preparing a revised version; details are in the note below.

  Abstract and notes

  [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 A_PL-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.

• Homotopy theory Preprint

  A recognition principle for iterated suspensions as coalgebras over the little cubes operad

  Oisín Flynn-Connolly, José M. Moreno-Fernández and Felix Wierstra

  Preprint, 2022

  arXivPoster

  Abstract and notes

  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.