arXiv subjects

Luciana Basualdo Bonatto

Publications and source records attributed to Luciana Basualdo Bonatto.

5 recordsLinked to original sources

arXiv2026

Galois actions on surfaces and a higher genus Grothendieck-Teichmüller group

We construct a modular operad $\mathbf{S}$ of mapping class groupoids modelling the higher-genus Teichmüller tower, with compositions and contractions encoding surface gluing. We prove a presentation theorem for maps out of $\mathbf{S}$ using genus-zero and genus-one generators and relations. Applying the work of Nakamura--Schneps, we obtain a faithful action of their subgroup $\widehatΓ\subseteq \widehat{\mathsf{GT}}$ on $\widehat{\mathbf{S}}$, and hence a faithful Galois action. The genus-zero truncation identifies with the cyclic operad of parenthesized ribbon braids, whose object-fixing profinite automorphism group is $\widehat{\mathsf{GT}}$. The completed classifying spaces form a modular $\infty$-operad whose corolla values are geometric étale homotopy types of moduli stacks of curves with nonzero tangent vectors at their marked points. The $\widehatΓ$-action induces a faithful action by homotopy automorphisms on the associated tower of profinite homotopy $1$-types, which agrees with the étale model in genus zero and one.

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arXiv2022

Decoupling generalised configuration spaces on surfaces

The configuration space of k points on a manifold carries an action of its diffeomorphism group. The homotopy quotient of this action is equivalent to the classifying space of diffeomorphisms of a punctured manifold, and therefore admits results about homological stability. Inspired by the works of Segal, McDuff, Bodigheimer, and Salvatore, we look at generalised configuration spaces where particles have labels and even partially summable labels, in which points are allowed to collide whenever their labels are summable. These generalised configuration spaces also admit actions of the diffeomorphism group and we look at their homotopy quotients. Our main result is a decoupling theorem for these homotopy quotients on surfaces: in a range, their homology is completely described by the product of the moduli space of surfaces and a generalised configuration space of points in $\mathbb{R}^\infty$. Using this result, we show these spaces admit homological stability with respect to increasing the genus, and we identify the stable homology. This can be interpreted as a Diff-equivariant homological stability for factorization homology. In addition, we use this result to study the group completion of the monoid of moduli spaces of configurations on surfaces.

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arXiv2021

The Influence of Memory in Multi-Agent Consensus

Multi-agent consensus problems can often be seen as a sequence of autonomous and independent local choices between a finite set of decision options, with each local choice undertaken simultaneously, and with a shared goal of achieving a global consensus state. Being able to estimate probabilities for the different outcomes and to predict how long it takes for a consensus to be formed, if ever, are core issues for such protocols. Little attention has been given to protocols in which agents can remember past or outdated states. In this paper, we propose a framework to study what we call \emph{memory consensus protocol}. We show that the employment of memory allows such processes to always converge, as well as, in some scenarios, such as cycles, converge faster. We provide a theoretical analysis of the probability of each option eventually winning such processes based on the initial opinions expressed by agents. Further, we perform experiments to investigate network topologies in which agents benefit from memory on the expected time needed for consensus.

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arXiv2020

Decoupling decorations on moduli spaces of manifolds

We consider moduli spaces of $d$-dimensional manifolds with embedded particles and discs. In this moduli space, the location of the particles and discs is constrained by the $d$-dimensional manifold. We will compare this moduli space with the moduli space of $d$-dimensional manifolds in which the location of such decorations is no longer constrained, i.e. the decorations are decoupled. We generalise work by Bödigheimer--Tillmann for oriented surfaces and obtain new results for surfaces with different tangential structures as well as to higher dimensional manifolds. We also provide a generalisation of this result to moduli spaces with more general submanifold decorations and specialise in the case of decorations being unparametrised unlinked circles.

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arXiv2020

An infinity operad of normalized cacti

We show that normalized cacti form an $\infty$-operad in the form of a dendroidal space satisfying a weak Segal condition. To do this, we introduce a new topological operad of bracketed trees and an enrichment of the dendroidal category $Ω$.

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