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Darío Sánchez Gómez

Publications and source records attributed to Darío Sánchez Gómez.

9 recordsLinked to original sources
arXiv2026

From Persistence to Resilience: New Betti Numbers for Analyzing Robustness in Simplicial Complex Networks

We show how persistence can be used to measure the robustness of cohomological cycles in finite simplicial complexes. We introduce two families of invariants: thick Betti numbers, which measure whether connected components and holes are supported by simplices of sufficiently high dimension; and cohesive Betti numbers, which measure the strength of higher-order adjacencies supporting cohomology classes. We then study robustness under simplicial degradation processes by combining an attack parameter with either the thick or the cohesive parameter, and by analyzing the resulting ladder modules through their image, kernel, and cokernel persistence modules. This allows us to distinguish features that remain structurally robust from those that lose thickness or cohesion during the degradation process. Finally, we prove stability results using the second network distance, which provides a theoretical reliability guarantee for the proposed constructions.

math.AT↗
arXiv2026

Interval Decompositions for Multipersistence Modules over Finite Posets and Robustness of Sheaf Data on Simplicial Complexes

We prove structure theorems for multipersistence modules indexed by finite posets that are not totally ordered. Specifically, we consider pointwise finite-dimensional modules over the opposite of the poset of non-empty subsets of a finite set, and give sufficient conditions, expressed through transition morphisms, for such modules to split as direct sums of interval modules. In the general case, the interval summands and multiplicities are explicitly determined by dimensions at finitely many indices. Although the assumptions may look algebraically restrictive, we show that they have a natural geometric origin in a robustness theory of cellular sheaf data over simplicial complexes, where one studies how algebraic information, compatibility constraints, and cohomological obstructions persist under structural failures. We extend thickness and cohesion from simplicial cohomology to cellular sheaves: thickness detects the dependence of cohomology classes on high-dimensional support, while cohesion captures the influence of higher-order adjacencies on the cohomological features. We leverage our abstract structure theorems to obtain interval decompositions for the resulting geometric cohesion modules. Finally, we introduce biparameter persistence constructions for sheaf resilience, tracking whether global sections and cohomological obstructions remain detectable on thick or cohesive substructures during topological degradation.

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arXiv2024

On Koszul complex of a supermodule

This paper is devoted to an exposition of the Koszul complex of a supermodule and its Berezinian from an intrinsic and as general as possible point of view. As an application, an analogue to Bott's formula in the supercommutative setting computing the cohomology of twisted differential p-forms sheaves on the projective superspaces is given.

math.AG↗
arXiv2020

Centrality measures in simplicial complexes: applications of Topological Data Analysis to Network Science

Many real networks in social sciences, biological and biomedical sciences or computer science have an inherent structure of simplicial complexes reflecting many-body interactions. Therefore, to analyse topological and dynamical properties of simplicial complex networks centrality measures for simplices need to be proposed. Many of the classical complex networks centralities are based on the degree of a node, so in order to define degree centrality measures for simplices (which would characterise the relevance of a simplicial community in a simplicial network), a different definition of adjacency between simplices is required. The aim of these notes is threefold: first we will use the recently introduced notions of higher order simplicial degrees to propose new degree based centrality measures in simplicial complexes. These theoretical centrality measures, such as the simplicial degree centrality or the eigenvector centrality would allow not only to study the relevance of a simplicial community and the quality of its higher-order connections in a simplicial network, but also they might help to elucidate topological and dynamical properties of simplicial networks; sencond, we define notions of walks and distances in simplicial complexes in order to study connectivity of simplicial networks and to generalise, to the simplicial case, the well known closeness and betweenness centralities (needed for instance to study the relevance of a simplicial community in terms of its ability of transmitting information); third, we propose a new clustering coefficient for simplices in a simplicial network, different from the one knows so far and which generalises the standard graph clustering of a vertex. This measure should be essential to know the density of a simplicial network in terms of its simplicial communities.

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arXiv2020

Simplicial degree in complex networks. Applications of Topological Data Analysis to Network Science

Network Science provides a universal formalism for modelling and studying complex systems based on pairwise interactions between agents. However, many real networks in the social, biological or computer sciences involve interactions among more than two agents, having thus an inherent structure of a simplicial complex. We propose new notions of higher-order degrees of adjacency for simplices in a simplicial complex, allowing any dimensional comparison among them and their faces, which as far as we know were lacked in the literature. We introduce multi-parameter boundary and coboundary operators in an oriented simplicial complex and also a novel multi-combinatorial Laplacian is defined, which generalises the graph and combinatorial Laplacian. To illustrate the potential applications of these theoretical results, we perform a structural analysis of higher-order connectivity in simplicial-complex networks by studying the associated distributions with these simplicial degrees in 17 real-world datasets coming from different domains such as coauthor networks, cosponsoring Congress bills, contacts in schools, drug abuse warning networks, e-mail networks or publications and users in online forums. We find rich and diverse higher-order connectivity structures and observe that datasets of the same type reflect similar higher-order collaboration patterns. Furthermore, we show that if we use what we have called the maximal simplicial degree (which counts the distinct maximal communities in which our simplex and all its strict sub-communities are contained), then its degree distribution is, in general, surprisingly different from the classical node degree distribution.

cs.SI↗
arXiv2015

Euler Sequence and Koszul complex of a module

We construct relative and global Euler sequences of a module. We apply it to prove some acyclicity results of the Koszul complex of a module and to compute the cohomology of the sheaves of (relative and absolute) differential $p$-forms of a projective bundle. In particular we generalize Bott's formula for the projective space to a projective bundle over a scheme of characteristic zero.

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arXiv2012

Relative Fourier-Mukai transforms for Weierstrass fibrations, abelian schemes and Fano fibrations

We study the group of relative Fourier-Mukai transforms for Weierstrass fibrations, abelian schemes and Fano or anti-Fano fibrations. For Weierstrass and Fano or anti-Fano fibrations we are able to describe this group completely. For abelian schemes over an arbitrary base we prove that if two of them are relative Fourier-Mukai partners then there is an isometric isomorphism between the fibre products of each of them and its dual abelian scheme. If the base is normal and the slope map is surjective we show that these two conditions are equivalent. Moreover in this situation we completely determine the group of relative Fourier-Mukai transforms and we prove that the number of relative Fourier-Mukai partners of a given abelian scheme over a normal base is finite.

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arXiv2009

Moduli Spaces of Semistable Sheaves on Singular Genus One Curves

We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli spaces of semistable sheaves on the curve. For rank zero, the moduli spaces are symmetric powers of the curve whilst for a fixed positive rank there are only a finite number of non-isomorphic spaces. We prove similar results for the relative semistable moduli spaces on an arbitrary genus one fibration with no conditions either on the base or on the total space. For a cycle $E_N$ of projective lines, we show that the unique degree 0 stable sheaves are the line bundles having degree 0 on every irreducible component and the sheaves $\mathcal{O}(-1)$ supported on one irreducible component. We also prove that the connected component of the moduli space that contains vector bundles of rank $r$ is isomorphic to the $r$-th symmetric product of the rational curve with one node.

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