Research
The papers and preprints below are organised by topic.
Homotopy theory
- A p-adic de Rham complex (ArXiv) Currently being corrected Abstract
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 (ArXiv) Algebraic and Geometric Topology (to appear)
Abstract
This is the first in a sequence of articles exploring the relationship between commutative algebras and E∞ algebras in characteristic p and mixed characteristic. In this paper we lay the groundwork by defining a new class of cohomology operations over Fp called cotriple products, generalising Massey products. We compute the secondary cohomology operations for a strictly commutative dg-algebra and the obstruction theories these induce, constructing several counterexamples to characteristic 0 behaviour, one of which answers a question of Campos, Petersen, Robert-Nicoud and Wierstra. We construct some families of higher cotriple products and comment on their behaviour. Finally, we distinguish a subclass of cotriple products that we call higher Steenrod operations and conclude with our main theorem, which says that E∞ algebras can be rectified if and only if the higher Steenrod operations vanish coherently.
- 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.
- 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.
Number Theory
- On the divisibility of sums of Fibonacci numbers (ArXiv), To appear in Integers: Electronic Journal of Combinatorial Number Theory 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.
Expository
- On associative and commutative differential graded algebras in positive characteristic (pdf), Submitted Abstract
In this expository piece, we construct an example in characteristic $p$ of two commutative dg-algebras which are quasi-isomorphic as associative but not commutative dg-algebras.
In preparation
These articles currently exist as the final two chapters of my thesis and will hopefully be adapted into articles shortly.
- Homotopically, E-infinity algebras do not generalise commutative dg-algebras Abstract
This is the third article exploring the relationship between commutative algebras and E∞-
algebras in characteristic p and mixed characteristic. In this paper, we show that, in characteristic
2, the homotopy category of strictly commutative dg-algebras does not form a subcategory of
the homotopy category of E∞-algebras. This is done by constructing a explicit example of two
strictly commutative algebras that are quasi-isomorphic in the category of E∞-algebras but not
the category of strictly commutative algebras. The construction is based on the theory of cotriple
operations.
- A higher Hochschild-Konstant-Rosenberg Theorem and the Deligne conjecture Abstract
We prove a generalisation of the Hochschild–Kostant–Rosenberg theorem that holds for all
commutative algebras and formal spaces. Using a A∞-version of the coendomorphism operad,
we define the notion of a coalgebra on the Hochschild homology. Finally, we specialise to the case
of iterated suspensions and construct the En+1-coalgebra coming from the Deligne conjecture
on a model for the Hochschild chain complex.
Ongoing projects
- A categorical approach to optimization (with Henning Basold, Chase Ford and Hao Wang) Abstract
Using the framework of enriched categories, we aim to study optimization processes, especially those with a homotopical component. To do this, we first need to get a good handle on stochastic differential equations, bringing categorical probability into play. To understand these, we need to have a good understanding of Gaussian processes. The first stage in this project therefore is to understand the Central Limit Theorem in the categorical context, something that also gives us a good handle on notions of convergence. After that, we aim to understand SDEs in the category of diffeological spaces. In the final stage of the project, homotopy will come into play.
