arXiv subjects

Jesus Gonzalez

Publications and source records attributed to Jesus Gonzalez.

At least 19 recordsLinked to original sources

arXiv2026

Underwriting the Agent Economy: The Blueprint for an AI Insurance Stack

From maritime trade to commercial nuclear power, insurance has been the enabler of major economic and technological developments by pricing risk, limiting downside, and spreading best practices. The emerging AI agent economy, projected to handle trillions of dollars in transactions by 2030, looks to be the next such development. Yet insurers' exposure to AI agent risk currently sits largely unpriced across existing insurance lines; between this silent coverage and growing exclusions, coverage is not fit for purpose. Furthermore, insurability is trending the wrong way: AI agent capabilities appear to be outpacing reliability, leading to rising incident severity; concentration among a few foundation model providers threatens correlated losses; and traditional actuarial modeling will struggle to keep pace with a technology evolving as rapidly as frontier AI. This report argues that affirmative AI coverage with limits in the billions is achievable by 2030, but only with industry-wide coordination. Drawing on successful historical precedents such as Underwriters Laboratories, the Closed Claims Project, and others, we lay out an eight-component AI insurance stack spanning incident data collection, catastrophe modeling, standards, contract design, risk selection, pricing, monitoring, and claims management. Building out this infrastructure is what will enable insurers to cover and manage AI agent risk sustainably and at scale. Finally, we discuss coverage for catastrophic risk from frontier AI ("AI CAT"), including CBRN, critical infrastructure collapse, and loss of control scenarios. Addressing these tail risks will require purpose-built instruments, potentially including a frontier model developer mutual, catastrophe bonds, bespoke liability regimes, and government backstops.

cs.CY↗

arXiv2023

An algorithmic discrete gradient field for non-colliding cell-like objects and the topology of pairs of points on skeleta of simplexes

For a positive integer $n$ and a finite simplicial complex $K$, we describe an algorithmic procedure constructing a maximal discrete gradient field $W(K,n)$ on Abrams' discretized configuration space $\text{DConf}(K,n)$. Computer experimentation shows that the field is generically optimal. We study the field $W(K,n)$ for $n=2$ and $K=\Delta^{m,d}$, the $d$-dimensional skeleton of the $m$-dimensional simplex. In particular, we prove that $\text{DConf}(\Delta^{m,d},2)$ is $(\min\{d,m-1\}-1)$-connected, has torsion-free homology and admits a minimal cell structure. We compute the Betti numbers of $\text{DConf}(\Delta^{m,d},2)$ and, for certain values of $d$, we prove that $\text{DConf}(\Delta^{m,d},2)$ breaks, up to homotopy, as a wedge of (not necessarily equidimensional) spheres.

math.AT↗

arXiv2021

Energy security: key concepts, components, and change of the paradigm

The authors of the study conduct a legal analysis of the concept of energy security. Energy is vital for sustainable development, and sustainability is not only at the heart of development, but also economic, environmental, social and military policies. To ensure the sustainability of the policy, 'security' seems to be a mandatory goal to achieve. The article critically assesses the change in the energy paradigm.

econ.GN↗

arXiv2021

A randomized greedy algorithm for piecewise linear motion planning

We describe and implement a randomized algorithm that inputs a polyhedron, thought of as the space of states of some automated guided vehicle $\mathcal{R}$, and outputs an explicit system of piecewise linear motion planners for $\mathcal{R}$. The algorithm is designed in such a way that the cardinality of the outputed system is probabilistically close (with parameters chosen by the user) to minimal possible. This yields the first automated solution for robust-to-noise robot motion planning in terms of simplicial complexity (SC) techniques, a discretization of Farber's topological complexity TC. Besides its relevance toward technological applications, our work revels that, unlike other discrete approaches to TC, the SC model can recast Farber's invariant without having to introduce costly subdivisions. We develop and implement our algorithm by actually discretizing Macías-Virgós and Mosquera-Lois' notion of homotopic distance, thus encompassing computer estimations of other sectional category invariants as well, such as the Lusternik-Schnirelmann category of polyhedra.

math.AT↗

arXiv2021

Multitasking collision-free motion planning algorithms in Euclidean spaces

We present optimal motion planning algorithms which can be used in designing practical systems controlling objects moving in Euclidean space without collisions. Our algorithms are optimal in a very concrete sense, namely, they have the minimal possible number of local planners. Our algorithms are motivated by those presented by Mas-Ku and Torres-Giese (as streamlined by Farber), and are developed within the more general context of the multitasking (a.k.a.~higher) motion planning problem. In addition, an eventual implementation of our algorithms is expected to work more efficiently than previous ones when applied to systems with a large number of moving objects.

cs.RO↗

arXiv2020

Right-angled Artin groups, polyhedral products and the TC-generating function

For a graph $Γ$, let $K(H_Γ,1)$ denote the Eilenberg-Mac Lane space associated to the right-angled Artin (RAA) group $H_Γ$ defined by $Γ$. We use the relationship between the combinatorics of $Γ$ and the topological complexity of $K(H_Γ,1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$, we construct a graph $\mathcal{O}_n$ whose TC-generating function has polynomial numerator of degree $n$. Additionally, motivated by the fact that $K(H_Γ,1)$ can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups.

math.AT↗

arXiv2018

The homotopy type of skeleta of the flag complex over a finite vector space

The aim of this paper is to give a (discrete) Morse theoretic proof of the fact that the $k$-th skeleton of the flag complex $\mathcal{F}$, associated to the lattice of subspaces of a finite dimensional vector space, is homotopy equivalent to a wedge of spheres of dimension $\min\{k,\dim(\mathcal{F})\}$. The tight control provided by Morse theoretic methods allows us to give an explicit formula for the number of spheres appearing in each of these wedge summands.

math.AT↗

arXiv2016

Projective product coverings and sequential motion planning algorithms in real projective spaces

For positive integers $m$ and $s$, let $\mathbf{m}_s$ stand for the $s$-th tuple $(m,\ldots,m)$. We show that, for large enough $s$, the higher topological complexity $TC_s$ of an even dimensional real projective space $RP^m$ is characterized as the smallest positive integer $k=k(m,s)$ for which there is a $(\mathbb{Z}_2)^{s-1}$-equivariant map from Davis' projective product space $P_{\mathbf{m}_s}$ to the $(k+1)$-th join-power $((\mathbb{Z}_2)^{s-1})^{\ast(k+1)}$. This is a (partial) generalization of Farber-Tabachnikov-Yuzvinsky's work relating $TC_2$ to the immersion dimension of real projective spaces. In addition, we compute the exact value of $TC_s(RP^m)$ for $m$ even and $s$ large enough.

math.AT↗

arXiv2014

Sequential motion planning of non-colliding particles in Euclidean spaces

In terms of Rudyak's generalization of Farber's topological complexity of the path motion planning problem in robotics, we give a complete description of the topological instabilities in any sequential motion planning algorithm for a system consisting of non-colliding autonomous entities performing tasks in space whilst avoiding collisions with several moving obstacles. The Isotopy Extension Theorem from manifold topology implies, somewhat surprisingly, that the complexity of this problem coincides with the complexity of the corresponding problem in which the obstacles are stationary.

math.AT↗

arXiv2013

Complex-projective and lens product spaces

Let $t$ be a positive integer. Following work of D. M. Davis, we study the topology of complex-projective product spaces, i.e. quotients of cartesian products of odd dimensional spheres by the diagonal $S^1$-action, and of the $t$-torsion lens product spaces, i.e. the corresponding quotients when the action is restricted to the $t^{\mathrm{th}}$ roots of unity. For a commutative complex-oriented cohomology theory $h^*$, we determine the $h^*$-cohomology ring of these spaces (in terms of the $t$-series for $h^*$, in the case of $t$-torsion lens product spaces). When $h^*$ is singular cohomology with mod 2 coefficients, we also determine the action of the Steenrod algebra. We show that these spaces break apart after a suspension as a wedge of desuspensions of usual stunted complex projective ($t$-torsion lens) spaces. We estimate the category and topological complexity of complex-projective and lens product spaces, showing in particular that these invariants are usually much lower than predicted by the usual dimensional bounds. We extend Davis' analysis of manifold properties such as immersion dimension, (stable-)span, and (stable-)parallelizability of real projective product spaces to the complex-projective and lens product cases.

math.AT↗

arXiv2013

Higher topological complexity and its symmetrization

We develop the properties of the $n$-th sequential topological complexity $TC_n$, a homotopy invariant introduced by the third author as an extension of Farber's topological model for studying the complexity of motion planning algorithms in robotics. We exhibit close connections of $TC_n(X)$ to the Lusternik-Schnirelmann category of cartesian powers of $X$, to the cup-length of the diagonal embedding $X\hookrightarrow X^n$, and to the ratio between homotopy dimension and connectivity of $X$. We fully compute the numerical value of $TC_n$ for products of spheres, closed 1-connected symplectic manifolds, and quaternionic projective spaces. Our study includes two symmetrized versions of $TC_n(X)$. The first one, unlike Farber-Grant's symmetric topological complexity, turns out to be a homotopy invariant of $X$; the second one is closely tied to the homotopical properties of the configuration space of cardinality-$n$ subsets of $X$. Special attention is given to the case of spheres.

math.AT↗

arXiv2013

Nuclear Activity In Isolated Galaxies

We present a spectroscopic study of the incidence of AGN nuclear activity in two samples of isolated galaxies (Karachentseva, V.E. & Varela, J.). Our results show that the incidence of non-thermal nuclear activity is about 43% and 31% for galaxies with emission lines and for the total sample 40% and 27% respectively. For the first time we have a large number of bona-fide isolated galaxies (513 objects), with statistically significant number of all types. We find a clear relation between bulge mass and the incidence of nuclear activity in the sample with emission lines. This relation becomes flatter when we take into account the complete sample with no emission line galaxies. A large fraction (~70%) of elliptical galaxies or early type spirals have an active galactic nucleus and ~70% of them are LINERs. Only 3% of the AGN show the presence of broad lines (a not a single one can be classified as type 1 AGN). This is a remarkable result which is completely at odds with the unified model even if we consider warped or clumpy tori. Finally, we interpret the large fraction of AGN in isolated galaxies as the result of secular evolution of their supermasive black holes.

astro-ph.CO↗

arXiv2012

Topological complexity of motion planning in projective product spaces

We study Farber's topological complexity (TC) of Davis' projective product spaces (PPS's). We show that, in many non-trivial instances, the TC of PPS's coming from at least two sphere factors is (much) lower than the dimension of the manifold. This is in high contrast with the known situation for (usual) real projective spaces for which, in fact, the Euclidean immersion dimension and TC are two facets of the same problem. Low TC-values have been observed for infinite families of non-simply connected spaces only for H-spaces, for finite complexes whose fundamental group has cohomological dimension not exceeding 2, and now in this work for infinite families of PPS's. We discuss general bounds for the TC (and the Lusternik-Schnirelmann category) of PPS's, and compute these invariants for specific families of such manifolds. Some of our methods involve the use of an equivariant version of TC. We also give a characterization of the Euclidean immersion dimension of PPS's through generalized concepts of axial maps and, alternatively, non-singular maps. This gives an explicit explanation of the known relationship between the generalized vector field problem and the Euclidean immersion problem for PPS's.

math.AT↗

arXiv2012

The integral cohomology groups of configuration spaces of pairs of points in real projective spaces

We compute the integral homology and cohomology groups of configuration spaces of two distinct points on a given real projective space. The explicit answer is related to the (known multiplicative structure in the) integral cohomology---with simple and twisted coefficients---of the dihedral group of order 8 (in the case of unordered configurations) and the elementary abelian 2-group of rank 2 (in the case of ordered configurations). As an application, we complete the computation of the symmetric topological complexity of real projective spaces of dimension 2^i + d for d=0,1,2.

math.AT↗

arXiv2011

Biequivariant Maps on Spheres and Topological Complexity of Lens Spaces

Weighted cup-length calculations in singular cohomology led Farber and Grant in 2008 to general lower bounds for the topological complexity of lens spaces. We replace singular cohomology by K-theory, and weighted cup-length arguments by considerations with biequivariant maps on spheres to improve on Farber-Grant's bounds by arbitrarily large amounts. Our calculations are based on the identification of key elements conjectured to generate the annihilator ideal of the toral bottom class in the ku-homology of the classifying space of a rank-2 abelian 2-group.

math.AT↗

arXiv2011

The integral cohomology of configuration spaces of pairs of points in real projective spaces

We compute the integral cohomology ring of configuration spaces of two points on a given real projective space. Apart from an integral class, the resulting ring is a quotient of the known integral cohomology of the dihedral group of order 8 (in the case of unordered configurations, thus has only 2- and 4-torsion) or of the elementary abelian 2-group of rank 2 (in the case of ordered configurations, thus has only 2-torsion). As an application, we complete the computation of the symmetric topological complexity of real projective spaces of dimensions of the form 2^i+j for non-negative i and j with j<3.

math.AT↗

arXiv2010

Symmetric topological complexity as the first obstruction in Goodwillie's Euclidean embedding tower for real projective spaces

As a first goal, it is explained why Goodwillie-Weiss calculus of embeddings offers new information about the Euclidean embedding dimension of P^m only for m < 16. Concrete scenarios are described in these low-dimensional cases, pinpointing where to look for potential, but critical, high-order obstructions in the corresponding Taylor towers. For m > 15, the relation TC^S(P^m) > n-1 is translated into the triviality of a certain cohomotopy Euler class which, in turn, becomes the only Taylor obstruction to producing an n-dimensional Euclidean embedding of P^m. A speculative bordism-type form of this primary obstruction is proposed as an analogue of Davis' BP-approach to the immersion problem of P^m. A form of the Euler class viewpoint is applied to show TC^S(P^3) = 5, as well as to suggest a few higher dimensional projective spaces for which the method could produce new information. As a second goal, the paper extends Farber's work on the motion planning problem in order to develop the notion of a symmetric motion planner for a mechanical system S. Following Farber's lead, this concept is connected to the symmetric topological complexity of the state space of S. The paper ends by sketching the construction of a concrete 5-local-rules symmetric motion planner for P^3.

math.AT↗

arXiv2009

On the embedding dimension of 2-torsion lens spaces

Using the $ku$- and $BP$-theoretic versions of Astey's cobordism obstruction for the existence of smooth Euclidean embeddings of stably almost complex manifolds, we prove that, for $e$ greater than or equal to $α(n)$--the number of ones in the dyadic expansion of $n$--, the ($2n+1$)-dimensional $2^e$-torsion lens space cannot be embedded in Euclidean space of dimension $4n-2α(n)+1$. A slightly restricted version of this fact holds for $e<α(n)$. We also give an inductive construction of Euclidean embeddings for $2^e$-torsion lens spaces. Some of our best embeddings are within one dimension of being optimal.

math.AT↗