53 citations · 99 across the 21 of their papers we have counts for
23 papers
The transverse singular complex
Greg Friedman, Anibal M. Medina-Mardones, Dev Sinha
Let be a smooth manifold without boundary and let be a countable collection of manifolds with corners, each equipped with a smooth map to . We show that the si…
Operadic Calculus for Higher Colour-Kinematics Duality
Anibal M. Medina-Mardones, Bruno Vallette
The search for algebraic foundations of colour-kinematics duality and the double-copy construction has brought into focus a generalization of Batalin--Vilkovisky algebras, referred…
Framed Polytopes and Higher Categories
Guillaume Laplante-Anfossi, Anibal M. Medina-Mardones, Arnau Padrol
In the early 1990s, Kapranov and Voevodsky proposed a geometric method for constructing higher-categorical pasting diagrams from generically framed convex polytopes. This work revi…
What makes math problems hard for reinforcement learning: a case study
Ali Shehper, Anibal M. Medina-Mardones, Lucas Fagan +6
Using a long-standing conjecture from combinatorial group theory, we explore, from multiple perspectives, the challenges of finding rare instances carrying disproportionately high…
Foundations of geometric cohomology: from co-orientations to product structures
Greg Friedman, Anibal M. Medina-Mardones, Dev Sinha
This manuscript develops a geometric approach to ordinary cohomology of smooth manifolds, constructing a cochain complex model based on co-oriented smooth maps from manifolds with…
Ranicki-Weiss assembly and the Steenrod construction
Anibal M. Medina-Mardones
We show that the Ranicki-Weiss assembly functor, going from chain complex valued presheaves on a simplicial complex to comodules over its Alexander-Whitney coalgebra, factors fully…
The diagonal of cellular spaces and effective algebro-homotopical constructions
Anibal M. Medina-Mardones
In this survey article we discuss certain homotopy coherent enhancements of the coalgebra structure on cellular chains defined by an approximation to the diagonal. Over the rationa…
Adams' cobar construction as a monoidal -coalgebra model of the based loop space
Anibal M. Medina-Mardones, Manuel Rivera
We prove that the classical map comparing Adams' cobar construction on the singular chains of a pointed space and the singular cubical chains on its based loop space is a quasi-iso…
Persistent homology for functionals
Ulrich Bauer, Anibal M. Medina-Mardones, Maximilian Schmahl
We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence d…
A combinatorial -algebra structure on cubical cochains and the Cartan-Serre map
Ralph M. Kaufmann, Anibal M. Medina-Mardones
Cubical cochains are equipped with an associative product, dual to the Serre diagonal, lifting the graded commutative structure in cohomology. In this work we introduce through exp…
Flowing from intersection product to cup product
Greg Friedman, Anibal M. Medina-Mardones, Dev Sinha
We use a vector field flow defined through a cubulation of a closed manifold to reconcile the partially defined commutative product on geometric cochains with the standard cup prod…
New formulas for cup- products and fast computation of Steenrod squares
Anibal M. Medina-Mardones
Operations on the cohomology of spaces are important tools enhancing the descriptive power of this computable invariant. For cohomology with mod 2 coefficients, Steenrod squares ar…
A computer algebra system for the study of commutativity up-to-coherent homotopies
Anibal M. Medina-Mardones
The Python package ComCH is a lightweight specialized computer algebra system that provides models for well known objects, the surjection and Barratt-Eccles operads, parameterizing…
Multisimplicial chains and configuration spaces
Anibal M. Medina-Mardones, Andrea Pizzi, Paolo Salvatore
This paper presents a generalization to multisimplicial sets of previously defined -coalgebra structures on the chains of simplicial and cubical sets. We focus on the sur…
Cochain level May-Steenrod operations
Ralph M. Kaufmann, Anibal M. Medina-Mardones
Steenrod defined in 1947 the Steenrod squares on the mod 2 cohomology of spaces using explicit cochain formulae for the cup- products; a family of coherent homotopies derived fr…
Hyperharmonic analysis for the study of high-order information-theoretic signals
Anibal M. Medina-Mardones, Fernando E. Rosas, Sebastián E. Rodríguez +1
Network representations often cannot fully account for the structural richness of complex systems spanning multiple levels of organisation. Recently proposed high-order information…
A Cochain Level Proof of Adem Relations in the Mod 2 Steenrod Algebra
Greg Brumfiel, Anibal M. Medina-Mardones, John Morgan
In 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-i products of cocycles. He later recast the cons…
giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration
Guillaume Tauzin, Umberto Lupo, Lewis Tunstall +6
We introduce giotto-tda, a Python library that integrates high-performance topological data analysis with machine learning via a scikit-learn-compatible API and state-of-the-art C+…
An algebraic representation of globular sets
A. M. Medina-Mardones
We describe a fully faithful embedding of the category of (reflexive) globular sets into the category of counital cosymmetric -coalgebras when is an integral domain. This em…
Persistence Steenrod modules
Umberto Lupo, Anibal M. Medina-Mardones, Guillaume Tauzin
It has long been envisioned that the strength of the barcode invariant of filtered cellular complexes could be increased using cohomology operations. Leveraging recent advances in…
An axiomatic characterization of Steenrod's cup- products
Anibal M. Medina-Mardones
We show that any natural construction of cup- products on the normalized cochains of simplicial sets, parameterized by the minimal free resolution of the trivial $\mathbb{F}_2[\…
A finitely presented -prop II: cellular context
Anibal M. Medina-Mardones
We construct, using finitely many generating cell and relations, props in the category of CW-complexes with the property that their associated operads are models for the …
A finitely presented -prop I: algebraic context
Anibal M. Medina-Mardones
We introduce a finitely presented prop in the category of differential graded modules whose associated operad $U(\mathcal{S})=\{\mathcal{S}(1,m…