Effective Computation of Generalized Spectral Sequences

Guidolin, Andrea; Romero, Ana

doi:10.1145/3208976.3208984

Cited by 6 publications

(17 citation statements)

References 12 publications

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“…S ) denote the terms of the associated spectral system. In [12] we stated the following result. Now, taking into account now the relations between multipersistence and spectral systems studied in Section 3, we obtain the following corollary.…”

Section: Computation Of a New Invariantmentioning

confidence: 93%

“…However, as in the case of spectral sequences associated with a linear filtration, no algorithm is provided to compute the different components when the initial chain complexes are not finitely generated. Thanks to the method of effective homology [27], in [12] we developed an algorithm for computing spectral systems of spaces (possibly) of infinite type. The corresponding programs were implemented as a new module for the system Kenzo [7], a symbolic computation software written in Common Lisp and devoted to algebraic topology, solving in this way also the classical problems of spectral sequences: determining differential maps and extensions.…”

Section: Spectral Systemsmentioning

confidence: 99%

“…Trying to generalize the existing programs, each of which was developed to deal with particular situations, we propose a new implementation of multipersistence as a new module for the system Kenzo, making use of our previous programs for computing spectral systems presented in [12]. Our new programs are in several respects more general than the existing ones, since they compute multipersistence over integer coefficients and they can be applied to filtrations over any poset.…”

Section: Generalizing the Rank Invariant In The Finite Casementioning

confidence: 99%

“…In our case, we have developed a Kenzo function computing the group with integer coefficients, producing not only the rank but also the generators and the torsion coefficients. Our programs use some previous functions computing spectral systems developed in [12], since some of the subgroups appearing in the quotient (4) are similar to the subgroups appearing in the spectral system terms (2). Once these subgroups are determined, the corresponding quotient can be computed by means of diagonalization algorithms of matrices in a similar way to the algorithm used to compute homology groups by means of the Smith Normal Form technique (see [14]).…”

Section: Generalizing the Rank Invariant In The Finite Casementioning

confidence: 99%

“…In this paper we clarify the relation between generalized persistent homology and spectral systems in a general scenario, and use it to adapt our programs for spectral systems implemented in the Kenzo system [12] to perform useful computations in the context of multipersistence. In addition to computing well-studied invariants of multipersistence in a new way, which differentiates in some key aspects from all the available implementations, we propose and compute a new invariant for multipersistence.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Computing Multipersistence by Means of Spectral Systems

Guidolin

Divasón

Romero

et al. 2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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In their original setting, both spectral sequences and persistent homology are algebraic topology tools defined from filtrations of objects (e.g. topological spaces or simplicial complexes) indexed over the set Z of integer numbers. Recently, generalizations of both concepts have been proposed which originate from a different choice of the set of indices of the filtration, producing the new notions of multipersistence and spectral system. In this paper, we show that these notions are related, generalizing results valid in the case of filtrations over Z. By using this relation and some previous programs for computing spectral systems, we have developed a new module for the Kenzo system computing multipersistence. We also present a new invariant providing information on multifiltrations and applications of our algorithms to spaces of infinite type.

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Section: Computation Of a New Invariantmentioning

confidence: 93%

Section: Spectral Systemsmentioning

confidence: 99%

Section: Generalizing the Rank Invariant In The Finite Casementioning

confidence: 99%

Section: Generalizing the Rank Invariant In The Finite Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computing Multipersistence by Means of Spectral Systems

Guidolin

Divasón

Romero

et al. 2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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Computing Higher Leray–Serre Spectral Sequences of Towers of Fibrations

Guidolin

Romero

2020

Found Comput Math

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The higher Leray-Serre spectral sequence associated with a tower of fibrations represents a generalization of the classical Leray-Serre spectral sequence of a fibration. In this work, we present algorithms to compute higher Leray-Serre spectral sequences leveraging the effective homology technique, which allows to perform computations involving chain complexes of infinite type associated with interesting objects in algebraic topology. In order to develop the programs, implemented as a new module for the Computer Algebra system Kenzo, we translated the original construction of the higher Leray-Serre spectral sequence in a simplicial framework and studied some of its fundamental properties.

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