Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio


IMPACT FACTOR 2018: 0.551

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

Mathematical Citation Quotient (MCQ) 2018: 0.27

Online
ISSN
1572-9176
See all formats and pricing
More options …
Volume 25, Issue 4

Issues

Algorithms in A -algebras

Mikael Vejdemo-JohanssonORCID iD: http://orcid.org/0000-0001-6322-7542
Published Online: 2018-10-05 | DOI: https://doi.org/10.1515/gmj-2018-0057

Abstract

Based on Kadeishvili’s original theorem inducing A-algebra structures on the homology of dg-algebras, several directions of algorithmic research in A-algebras have been pursued. In this paper, we survey the work done on calculating explicit A-algebra structures from homotopy retractions, in group cohomology and in persistent homology.

Keywords: perturbation theory; group cohomology; persistent homology

MSC 2010: 20J06; 55P48

Dedicated to Tornike Kadeishvili

References

  • [1]

    F. Belchí, Optimising the topological information of the A-persistence groups, preprint (2017), https://arxiv.org/abs/1706.06019.

  • [2]

    F. Belchí and A. Murillo, A-persistence, Appl. Algebra Engrg. Comm. Comput. 26 (2015), no. 1–2, 121–139. Google Scholar

  • [3]

    A. Berciano, Cálculo simbólico y técnicas de control de A-infinito estructuras Ph.D. thesis, Universidad de Sevilla, 2006. Google Scholar

  • [4]

    A. Berciano, H. Molina-Abril and P. Real, Searching high order invariants in computer imagery, Appl. Algebra Engrg. Comm. Comput. 23 (2012), no. 1–2, 17–28. CrossrefGoogle Scholar

  • [5]

    A. Berciano and P. Real, A-coalgebra structure on the p-homology of Eilenberg–Mac Lane spaces, Proceedings EACA (2004), http://www.ehu.eus/aba/articles/ainhoa-eaca04.pdf.

  • [6]

    A. Berciano and P. Real, A-coalgebra structure maps that vanish on H*(K(π,n);p), Forum Math. 22 (2010), no. 2, 357–378. Google Scholar

  • [7]

    A. Berciano and R. Umble, Some naturally occurring examples of A-bialgebras, J. Pure Appl. Algebra 215 (2011), no. 6, 1284–1291. Google Scholar

  • [8]

    A. Berciano-Alcaraz, A computational approach of A-(co)algebras, Int. J. Comput. Math. 87 (2010), no. 4, 935–953. Google Scholar

  • [9]

    W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3–4, 235–265. CrossrefGoogle Scholar

  • [10]

    R. Brown, The twisted Eilenberg–Zilber theorem, Simposio di Topologia (Messina 1964), Edizioni Oderisi, Gubbio (1965), 33–37. Google Scholar

  • [11]

    G. Carlsson, Topology and data, Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 2, 255–308. CrossrefGoogle Scholar

  • [12]

    F. Chazal, V. De Silva, M. Glisse and S. Oudot, The Structure and Stability of Persistence Modules, Springer Briefs Math., Springer, Cham, 2016. Google Scholar

  • [13]

    D. Cohen-Steiner, H. Edelsbrunner and J. Harer, Stability of persistence diagrams, Discrete Comput. Geom. 37 (2007), no. 1, 103–120. Web of ScienceCrossrefGoogle Scholar

  • [14]

    R. Ghrist, Barcodes: The persistent topology of data, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 1, 61–75. Google Scholar

  • [15]

    V. K. A. M. Gugenheim, L. A. Lambe and J. D. Stasheff, Perturbation theory in differential homological algebra. II, Illinois J. Math. 35 (1991), no. 3, 357–373. Google Scholar

  • [16]

    E. Herscovich, A higher homotopic extension of persistent (co) homology, preprint (2014), https://arxiv.org/abs/1412.1871.

  • [17]

    T. V. Kadeishvili, On the theory of homology of fiber spaces (in Russian), Uspekhi Mat. Nauk 35 (1980), no. 3(213), 183–188; translation in Russian Math. Surveys 35 (1980), no. 3, 231–238. Google Scholar

  • [18]

    B. Keller, Introduction to A-infinity algebras and modules, Homology Homotopy Appl. 3 (2001), no. 1, 1–35. CrossrefGoogle Scholar

  • [19]

    B. Keller, A-infinity algebras in representation theory, Representations of Algebra. Vol. I, II, Beijing Normal University Press, Beijing (2002), 74–86. Google Scholar

  • [20]

    D.-M. Lu, J. H. Palmieri, Q.-S. Wu and J. J. Zhang, A-algebras for ring theorists, Algebra Colloq. 11 (2004), no. 1, 91–128. Google Scholar

  • [21]

    D.-M. Lu, J. H. Palmieri, Q.-S. Wu and J. J. Zhang, A-infinity structure on Ext-algebras, preprint (2006), https://arxiv.org/abs/math/0606144.

  • [22]

    D. Madsen, Homological aspects in representation theory, Ph.D. thesis, Norges Teknisk-Naturvitenskapelige Universitet, 2002. Google Scholar

  • [23]

    S. A. Merkulov, Strong homotopy algebras of a Kähler manifold, Int. Math. Res. Not. IMRN 1999 (1999), no. 3, 153–164. CrossrefGoogle Scholar

  • [24]

    S. Saneblidze and R. Umble, Diagonals on the permutahedra, multiplihedra and associahedra, Homology Homotopy Appl. 6 (2004), no. 1, 363–411. CrossrefGoogle Scholar

  • [25]

    S. Schmid, An A-structure on the cohomology ring of the symmetric group Sp with coefficients in 𝔽p, Algebr. Represent. Theory 17 (2014), no. 5, 1553–1585. Google Scholar

  • [26]

    J. D. Stasheff, Homotopy associativity of H-spaces. I, Trans. Amer. Math. Soc. 108 (1963), 275–292. Google Scholar

  • [27]

    J. D. Stasheff, Homotopy associativity of H-spaces. II, Trans. Amer. Math. Soc. 108 (1963), 293–312. Google Scholar

  • [28]

    M. Vejdemo-Johansson, Enumerating the Saneblidze–Umble diagonal terms, preprint (2007), https://arxiv.org/abs/0707.4399.

  • [29]

    M. Vejdemo-Johansson, Computation of A-infinity algebras in group cohomology, Ph.D. thesis, Friedrich-Schiller-Universität Jena, 2008. Google Scholar

  • [30]

    M. Vejdemo-Johansson, A partial A-structure on the cohomology of Cn×Cm, J. Homotopy Relat. Struct. 3 (2008), no. 1, 1–11. Google Scholar

  • [31]

    M. Vejdemo-Johansson, Blackbox computation of A-algebras, Georgian Math. J. 17 (2010), no. 2, 391–404. Google Scholar

  • [32]

    M. Vejdemo-Johansson, Sketches of a platypus: A survey of persistent homology and its algebraic foundations, Algebraic Topology: Applications and new Directions, Contemp. Math. 620, American Mathematical Society, Providence (2014), 295–319. Google Scholar

  • [33]

    The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.8, 2017.

About the article

Received: 2017-11-15

Accepted: 2018-06-12

Published Online: 2018-10-05

Published in Print: 2018-12-01


Citation Information: Georgian Mathematical Journal, Volume 25, Issue 4, Pages 629–635, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2018-0057.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in