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Andrea Bianchi

Publications and source records attributed to Andrea Bianchi.

36 records · Page 2Linked to original sources
arXiv2024

Deloopings of Hurwitz spaces

For a partially multiplicative quandle (PMQ) $\mathcal{Q}$ we consider the topological monoid $\mathring{\mathrm{HM}}(\mathcal{Q})$ of Hurwitz spaces of configurations in the plane with local monodromies in $\mathcal{Q}$. We compute the group completion of $\mathring{\mathrm{HM}}(\mathcal{Q})$: it is the product of the (discrete) enveloping group $\mathcal{G}(\mathcal{Q})$ with a component of the double loop space of the relative Hurwitz space $\mathrm{Hur}_+([0,1]^2,\partial[0,1]^2;\mathcal{Q},G)_{1\!\!\,1}$; here $G$ is any group giving rise, together with $\mathcal{Q}$, to a PMQ-group pair. Assuming further that $\mathcal{Q}$ is finite and rationally Poincare and that $G$ is finite, we compute the rational cohomology ring of $\mathrm{Hur}_+([0,1]^2,\partial[0,1]^2;\mathcal{Q},G)_{1\!\!\,1}$.

math.AT↗
arXiv2024

Understanding and Shaping Human-Technology Assemblages in the Age of Generative AI

Generative AI capabilities are rapidly transforming how we perceive, interact with, and relate to machines. This one-day workshop invites HCI researchers, designers, and practitioners to imaginatively inhabit and explore the possible futures that might emerge from humans combining generative AI capabilities into everyday technologies at massive scale. Workshop participants will craft stories, visualisations, and prototypes through scenario-based design to investigate these possible futures, resulting in the production of an open-annotated scenario library and a journal or interactions article to disseminate the findings. We aim to gather the DIS community knowledge to explore, understand and shape the relations this new interaction paradigm is forging between humans, their technologies and the environment in safe, sustainable, enriching, and responsible ways.

cs.HC↗
arXiv2024

Theory of magnetic properties in QED environments: application to molecular aromaticity

In this work, we present ab initio cavity QED methods which include interactions with a static magnetic field and nuclear spin degrees of freedom using different treatments of the quantum electromagnetic field. We derive explicit expressions for QED-HF magnetizability, nuclear shielding, and spin-spin coupling tensors. We apply this theory to explore the influence of the cavity field on the magnetizability of saturated, unsaturated, and aromatic hydrocarbons, showing the effects of different polarization orientations and coupling strengths. We also examine how the cavity affects aromaticity descriptors, such as the nucleus-independent chemical shift and magnetizability exaltation. We employ these descriptors to study the trimerization reaction of acetylene to benzene. We show how the optical cavity induces modifications in the aromatic character of the transition state leading to variations in the activation energy of the reaction. Our findings shed light on the effects induced by the cavity on magnetic properties, especially in the context of aromatic molecules, providing valuable insights into understanding the interplay between the quantum electromagnetic field and molecules.

physics.chem-ph↗
arXiv2024

On two quotients of $S^2\times S^2$

In this note we prove that two seemingly different smooth 4-manifolds arising as quotients of $S^2\times S^2$ by free actions of $\mathbb{Z}/4$ are in fact diffeomorphic, answering a question of Hambleton and Hillman.

math.GT↗
arXiv2023

Hurwitz-Ran spaces

Given a couple of subspaces $\mathcal{Y}\subset\mathcal{X}$ of the complex plane $\mathbb{C}$ satisfying some mild conditions (a ``nice couple''), and given a PMQ-pair $(\mathcal{Q},G)$, consisting of a partially multiplicative quandle (PMQ) $\mathcal{Q}$ and a group $G$, we introduce a ``Hurwitz-Ran'' space $\mathrm{Hur}(\mathcal{X},\mathcal{Y};\mathcal{Q},G)$, containing configurations of points in $\mathcal{X}\setminus\mathcal{Y}$ and in $\mathcal{Y}$ with monodromies in $\mathcal{Q}$ and in $G$, respectively. We further introduce a notion of morphisms between nice couples, and prove that Hurwitz-Ran spaces are functorial both in the nice couple and in the PMQ-group pair. For a locally finite PMQ $\mathcal{Q}$ we prove a homeomorphism between $\mathrm{Hur}((0,1)^2;\mathcal{Q}_+)$ and the simplicial Hurwitz space $\mathrm{Hur}^Δ(\mathcal{Q})$, introduced in previous work of the author: this provides in particular $\mathrm{Hur}((0,1)^2;\mathcal{Q}_+)$ with a cell stratification in the spirit of Fox-Neuwirth and Fuchs.

math.AT↗
arXiv2023

Moduli spaces of Riemann surfaces as Hurwitz spaces

We consider the moduli space $\mathfrak{M}_{g,n}$ of Riemann surfaces of genus $g\ge0$ with $n\ge1$ ordered and directed marked points. For $d\ge 2g+n-1$ we show that $\mathfrak{M}_{g,n}$ is homotopy equivalent to a component of the simplicial Hurwitz space $\mathrm{Hur}^Δ(\mathfrak{S}_d^{\mathrm{geo}})$ associated with the partially multiplicative quandle $\mathfrak{S}_d^{\mathrm{geo}}$. As an application, we give a new proof of the Mumford conjecture on the stable rational cohomology of moduli spaces of Riemann surfaces. We also provide a combinatorial model for the infinite loop space $Ω^{\infty-2}\mathrm{MTSO}(2)$ of Hurwitz flavour.

math.AT↗
arXiv2023

On Generators and Relations of the Rational Cohomology of Hilbert Schemes

We consider for $d\geq 1$ the graded commutative $\mathbb{Q}$-algebra $\mathcal{A}(d):=H^*(\operatorname{Hilb}^d(\mathbb{C}^2);\mathbb{Q})$, which is also connected to the study of generalised Hurwitz spaces by work of the first author. These Hurwitz spaces are in turn related to the moduli spaces of Riemann surfaces with boundary. We determine two distinct, minimal sets of $\lfloor d/2\rfloor$ multiplicative generators of $\mathcal{A}(d)$. Additionally, we prove when the lowest degree generating relations occur. For small values of $d$ we also determine a minimal set of generating relations, which leads to several conjectures about the necessary generating relations for $\mathcal{A}(d)$.

math.AC↗
arXiv2023

A Decision Tree to Shepherd Scientists through Data Retrievability

Reproducibility is a crucial aspect of scientific research that involves the ability to independently replicate experimental results by analysing the same data or repeating the same experiment. Over the years, many works have been proposed to make the results of the experiments actually reproducible. However, very few address the importance of data reproducibility, defined as the ability of independent researchers to retain the same dataset used as input for experimentation. Properly addressing the problem of data reproducibility is crucial because often just providing a link to the data is not enough to make the results reproducible. In fact, also proper metadata (e.g., preprocessing instruction) must be provided to make a dataset fully reproducible. In this work, our aim is to fill this gap by proposing a decision tree to sheperd researchers through the reproducibility of their datasets. In particular, this decision tree guides researchers through identifying if the dataset is actually reproducible and if additional metadata (i.e., additional resources needed to reproduce the data) must also be provided. This decision tree will be the foundation of a future application that will automate the data reproduction process by automatically providing the necessary metadata based on the particular context (e.g., data availability, data preprocessing, and so on). It is worth noting that, in this paper, we detail the steps to make a dataset retrievable, while we will detail other crucial aspects for reproducibility (e.g., dataset documentation) in future works.

cs.DL↗
arXiv2022

Non-trivial action of the Johnson filtration on the homology of configuration spaces

We let the mapping class group $Γ_{g,1}$ of a genus $g$ surface $Σ_{g,1}$ with one boundary component act on the homology $H_*(F_{n}(Σ_{g,1});\mathbb{Q})$ of the $n^{th}$ ordered configuration space of the surface. We prove that the action is non-trivial when restricted to the $(n-1)^{st}$ stage of the Johnson filtration, for all $n\ge 1$ and $g\ge 2$. We deduce an analogous result for closed surfaces.

math.GT↗
arXiv2022

Parametrised moduli spaces of surfaces as infinite loop spaces

We study the $E_2$-algebra $Λ\mathfrak{M}_{*,1}=\coprod_{g\geqslant 0}Λ\mathfrak{M}_{g,1}$ consisting of free loop spaces of moduli spaces of Riemann surfaces with one parametrised boundary component, and compute the homotopy type of the group completion $ΩBΛ\mathfrak{M}_{*,1}$: it is the product of $Ω^\infty\mathbf{MTSO}(2)$ with a certain free $Ω^\infty$-space depending on the family of all boundary-irreducible mapping classes in all mapping class groups $Γ_{g,n}$ with $g\geqslant 0$ and $n\geqslant 1$.

math.AT↗
arXiv2022

Mapping class group actions on configuration spaces and the Johnson filtration

Let $F_n(Σ_{g,1})$ denote the configuration space of $n$ ordered points on the surface $Σ_{g,1}$ and let $Γ_{g,1}$ denote the mapping class group of $Σ_{g,1}$. We prove that the action of $Γ_{g,1}$ on $H_i(F_n(Σ_{g,1});\mathbb{Z})$ is trivial when restricted to the $i^{th}$ stage of the Johnson filtration $\mathcal{J}(i)\subset Γ_{g,1}$. We give examples showing that $\mathcal{J}(2)$ acts nontrivially on $H_3(F_3(Σ_{g,1}))$ for $g\ge 2$, and provide two new conceptual reinterpretations of a certain group introduced by Moriyama.

math.GT↗
arXiv2022

Vertical configuration spaces and their homology

We introduce ordered and unordered configuration spaces of 'clusters' of points in an Euclidean space $\mathbb{R}^d$, where points in each cluster satisfy a 'verticality' condition, depending on a decomposition $d=p+q$. We compute the homology in the ordered case and prove homological stability in the unordered case.

math.AT↗
arXiv2021

On the homology of the commutator subgroup of the pure braid group

We study the homology of $[P_n,P_n]$, the commutator subgroup of the pure braid group on $n$ strands, and show that $H_l([P_n,P_n])$ contains a free abelian group of infinite rank for all $1\leq l\leq n-2$. As a consequence we determine the cohomological dimension of $[P_n,P_n]$: for $n\geq 2$ we have $\mathrm{cd}([P_n,P_n])=n-2$.

math.AT↗
arXiv2020

Splitting of the homology of the punctured mapping class group

Let $Γ_{g,1}^m$ be the mapping class group of the orientable surface $Σ_{g,1}^m$ of genus $g$ with one parametrised boundary curve and $m$ permutable punctures; when $m=0$ we omit it from the notation. Let $β_{m}(Σ_{g,1})$ be the braid group on $m$ strands of the surface $Σ_{g,1}$. We prove that $H_*(Γ_{g,1}^m;\mathbb{Z}_2)\cong H_*(Γ_{g,1};H_*(β_{m}(Σ_{g,1});\mathbb{Z}_2))$. The main ingredient is the computation of $H_*(β_{m}(Σ_{g,1});\mathbb{Z}_2)$ as a symplectic representation of $Γ_{g,1}$.

math.AT↗
arXiv2019

Braid groups, mapping class groups and their homology with twisted coefficients

We consider the Birman-Hilden inclusion $φ\colon\mathfrak{Br}_{2g+1}\toΓ_{g,1}$ of the braid group into the mapping class group of an orientable surface with boundary, and prove that $φ$ is stably trivial in homology with twisted coefficients in the symplectic representation $H_1(Σ_{g,1})$ of the mapping class group; this generalises a result of Song and Tillmann regarding homology with constant coefficients. Furthermore we show that the stable homology of the braid group with coefficients in $φ^*(H_1(Σ_{g,1}))$ has only $4$-torsion.

math.AT↗
arXiv2018

Topological complexity of unordered configuration spaces of surfaces

We determine the topological complexity of unordered configuration spaces on almost all punctured surfaces (both orientable and non-orientable). We also give improved bounds for the topological complexity of unordered configuration spaces on all aspherical closed surfaces, reducing it to three possible values. The main methods used in the proofs were developed in 2015 by Grant, Lupton and Oprea to give bounds for the topological complexity of aspherical spaces. As such this paper is also part of the current effort to study the topological complexity of aspherical spaces and it presents many further examples where these methods strongly improve upon the lower bounds given by zero-divisor cup-length.

math.AT↗
arXiv2018

RealPen: Providing Realism in Handwriting Tasks on Touch Surfaces using Auditory-Tactile Feedback

We present RealPen, an augmented stylus for capacitive tablet screens that recreates the physical sensation of writing on paper with a pencil, ball-point pen or marker pen. The aim is to create a more engaging experience when writing on touch surfaces, such as screens of tablet computers. This is achieved by re-generating the friction-induced oscillation and sound of a real writing tool in contact with paper. To generate realistic tactile feedback, our algorithm analyses the frequency spectrum of the friction oscillation generated when writing with traditional tools, extracts principal frequencies, and uses the actuator's frequency response profile for an adjustment weighting function. We enhance the realism by providing the sound feedback aligned with the writing pressure and speed. Furthermore, we investigated the effects of superposition and fluctuation of several frequencies on human tactile perception, evaluated the performance of RealPen, and characterized users' perception and preference of each feedback type.

cs.HC↗