Darío Sánchez Gómez

• PhD
• University of Salamanca

Persistent homology is a fundamental tool in topological data analysis; however, it lacks methods to quantify the fragility or fineness of cycles, anticipate their formation or disappearance, or evaluate their stability beyond persistence. Furthermore, classical Betti numbers fail to capture key structural properties such as simplicial dimensions a...

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 com...

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. The relevance of an agent in a...

Many real networks in social sciences, biological and biomedical sciences or computer science have an inherent structure of simplicial complexes reflecting many-body interactions. Using the recently introduced higher order notions of adjacency and degree for simplices in a simplicial complex, we define new centrality measures in simplicial complexe...

Many real networks in social, biological or computer sciences have an inherent structure of a simplicial complex, which reflects the multi interactions among agents (and groups of agents) and constitutes the basics of Topological Data Analysis. Normally, the relevance of an agent in a network of graphs is given in terms of the number of edges incid...

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...

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 i...

We construct stable sheaves over K3 fibrations using a relative Fourier-Mukai transform which describes the sheaves in terms of spectral data similar to the construction for elliptic fibrations. On K3 fibered Calabi–Yau threefolds we show that the Fourier-Mukai transform induces an embedding of the relative Jacobian of spectral line bundles on spec...

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...

We construct stable sheaves over K3 fibrations using a relative Fourier-Mukai transform which describes the sheaves in terms of spectral data. This procedure is similar to the construction for elliptic fibrations, which we also describe. On K3 fibered Calabi-Yau threefolds, we show that the Fourier-Mukai transform induces an embedding of the relati...

Based on the Matsusaka-Ran criterion we give a criterion to characterize when a principal polarized abelian variety is a Jacobian by the existence of Picard bundles.

Based on the Matsusaka-Ran criterion we give a criterion to characterize when a principal polarized abelian variety is a Jacobian by the existence of Picard bundles.