Effective computation of invariants of finite topological spaces

ilustraciones, gráficas, tablas

Autores:: Cuevas Rozo, Julián Leonardo

Tipo de recurso:: Doctoral thesis

Fecha de publicación:: 2021

Institución:: Universidad Nacional de Colombia

Repositorio:: Universidad Nacional de Colombia

Idioma:: eng

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dc.title.eng.fl_str_mv	Effective computation of invariants of finite topological spaces
dc.title.translated.spa.fl_str_mv	Cálculo efectivo de invariantes de espacios topológicos finitos
title	Effective computation of invariants of finite topological spaces
spellingShingle	Effective computation of invariants of finite topological spaces 510 - Matemáticas::514 - Topología Topological spaces Invariants Vector analysis Espacios topológicos Invariantes Análisis vectorial Homology groups Homotopy invariants H-regular finite spaces Effective algorithms Weak homotopy types Computational topology Discrete vector fields Espacios topológicos finitos Grupos de homología Campos de vectores discretos Invariantes homotópicos Espacios finitos h-regulares Algoritmos efectivos Tipos de homotopía débil Topología computacional
title_short	Effective computation of invariants of finite topological spaces
title_full	Effective computation of invariants of finite topological spaces
title_fullStr	Effective computation of invariants of finite topological spaces
title_full_unstemmed	Effective computation of invariants of finite topological spaces
title_sort	Effective computation of invariants of finite topological spaces
dc.creator.fl_str_mv	Cuevas Rozo, Julián Leonardo
dc.contributor.advisor.spa.fl_str_mv	Lambán Pardo, Laureano Romero Ibáñez, Ana Sarria Zapata, Humberto
dc.contributor.author.spa.fl_str_mv	Cuevas Rozo, Julián Leonardo
dc.subject.ddc.spa.fl_str_mv	510 - Matemáticas::514 - Topología
topic	510 - Matemáticas::514 - Topología Topological spaces Invariants Vector analysis Espacios topológicos Invariantes Análisis vectorial Homology groups Homotopy invariants H-regular finite spaces Effective algorithms Weak homotopy types Computational topology Discrete vector fields Espacios topológicos finitos Grupos de homología Campos de vectores discretos Invariantes homotópicos Espacios finitos h-regulares Algoritmos efectivos Tipos de homotopía débil Topología computacional
dc.subject.lemb.eng.fl_str_mv	Topological spaces Invariants Vector analysis
dc.subject.lemb.spa.fl_str_mv	Espacios topológicos Invariantes Análisis vectorial
dc.subject.proposal.eng.fl_str_mv	Homology groups Homotopy invariants H-regular finite spaces Effective algorithms Weak homotopy types Computational topology Discrete vector fields
dc.subject.proposal.spa.fl_str_mv	Espacios topológicos finitos Grupos de homología Campos de vectores discretos Invariantes homotópicos Espacios finitos h-regulares Algoritmos efectivos Tipos de homotopía débil Topología computacional
description	ilustraciones, gráficas, tablas
publishDate	2021
dc.date.accessioned.none.fl_str_mv	2021-10-13T16:18:33Z
dc.date.available.none.fl_str_mv	2021-10-13T16:18:33Z
dc.date.issued.none.fl_str_mv	2021
dc.type.spa.fl_str_mv	Trabajo de grado - Doctorado
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dc.identifier.instname.spa.fl_str_mv	Universidad Nacional de Colombia
dc.identifier.reponame.spa.fl_str_mv	Repositorio Institucional Universidad Nacional de Colombia
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identifier_str_mv	Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia
dc.language.iso.spa.fl_str_mv	eng
language	eng
dc.relation.references.spa.fl_str_mv	J. J. Andrews and M. L. Curtis, Free groups and handlebodies, Proceedings of the American Mathematical Society, American Mathematical Society 16 (1965), no. 2, 192–195. P. Alexandroff, Diskrete R¨aume, Mat. Sb. (N.S.) 2 (1937), 501–518. J. A. Barmak, Algebraic topology of finite topological spaces and applications, Lecture Notes in Mathematics, vol. 2032, Springer, 2011. B. Benedetti and F. H. Lutz, Library of triangulations, http: //page.math.tu-berlin.de/˜lutz/stellar/library_ of_triangulations/, 2017. Y. Bengio, A. Lodi, and A. Prouvost, Machine learning for combinatorial optimization: a methodological tour d’horizon, https://arxiv.org/ pdf/1811.06128.pdf, 2020. N. Cianci and M. Ottina, A new spectral sequence for homology of posets, Topology and its Applications 217 (2017), 1–19. N. Cianci and M. Ottina, Poset splitting and minimality of finite models, Journal of Combinatorial Theory, Series A 157 (2018), 120–161. J. Cuevas-Rozo, Funciones submodulares y matrices en el estudio de los espacios topológicos finitos, Tesis de maestría, Universidad Nacional de Colombia, 2016. J. Cuevas-Rozo, Point reduction algorithms and discrete vector fields for finite topological spaces in the Kenzo system, Proceedings of Fourth EACA International School on Computer Algebra and its Applications, 2018, https://www.usc.es/regaca/eacaschool18/files/ Contributed_talks_EACA2018.pdf, pp. 3–4. J. Cuevas-Rozo, Cálculo de invariantes topológicos sobre espacios topológicos finitos, XXII Congreso Colombiano de Matemáticas 2019, 2019, http://scm.org.co/wp-content/uploads/2019/06/ Cronograma_XXII_CCM2019_V2.pdf. J. Cuevas-Rozo, h-regularización de espacios topológicos finitos, VIII Encuentro de Jóvenes Topólogos, 2019, http://xtsunxet.usc.es/etop2019/ EJTop2019-JCuevas.pdf. J. Cuevas-Rozo, Finite topological spaces in Kenzo, https://github.com/ jcuevas-rozo/finite-topological-spaces, 2020. J. Cuevas-Rozo, J. Divasón, M. Marco-Buzunáriz, and A. Romero, A Kenzo interface for algebraic topology computations in SageMath, ACM Commun. Comput. Algebra 53 (2019), no. 2, 61–64. J. Cuevas-Rozo, J. Divasón, M. Marco-Buzunáriz, and A. Romero, Integration of the Kenzo system within SageMath for new algebraic topology computations, Mathematics, 9(7), 722 (2021). J. Cuevas-Rozo, L. Lambán, A. Romero, and H. Sarria, New algorithms for computing homology of finite topological spaces, Proceedings of 24th Conference on Applications of Computer Algebra - ACA 2018, 2018, pp. 108– 112. J. Cuevas-Rozo, L. Lambán, A. Romero, and H. Sarria, Effective homological computations on finite topological spaces, Applicable Algebra in Engineering, Communication and Computing (2020), In press. J. Cuevas-Rozo, M. Marco-Buzunáriz, and A. Romero, Computing with Kenzo from Sage, Proceedings of the 2019 conference on Effective Methods in Algebraic Geometry, 2019, http://eventos.ucm.es/_files/_event/_12097/ _editorFiles/file/Abstracts_MEGA-alphabetic.pdf, p. 23. The SageMath Developers, The Sage Mathematics Software System (Version 9.2), https://www.sagemath.org, 2020. X. Dousson, J. Rubio, F. Sergeraert, and Y. Siret, The Kenzo program, Institut Fourier, Grenoble, 1999, http://www-fourier. ujf-grenoble.fr/˜sergerar/Kenzo/. X. Fernández, Finite-spaces, https://github.com/ ximenafernandez/Finite-Spaces, 2017. X. Fernández, Métodos combinatorios y algoritmos en topología de dimensiones bajas y la conjetura de Andrews-Curtis, Ph.D. thesis, Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales, 2017. X. Fernández, 3-deformations of 2-complexes and Morse theory, https:// arxiv.org/pdf/1912.00115.pdf, 2019. R. Forman, Morse theory for cell complexes, Advances in Mathematics 134 (1998), 90–145. A. Hatcher, Algebraic topology, Cambridge University Press, 2002. G. Heber, A repackaged version of the Kenzo program by Francis Sergeraert and collaborators, https://github.com/gheber/kenzo, 2019. J. Heras, A Kenzo module computing homotopy groups, 2011, https://esus.unirioja.es/psycotrip/archivos_ documentos/homotopy.cl. M.W. Jahn, P. E. Bradley, M. Al-Doori, and M. Breunig, Topologically consistent models for efficient big geo-spatio-temporal data distribution, ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences IV-4/W5 (2017), 65–72. A. I. Khan and S. Al-Habsi, Machine Learning in Computer Vision, Procedia Computer Science 167 (2020), 1444–1451. V. A. Kovalevsky, Finite topology as applied to image analysis, Computer Vision, Graphics and Image Processing 46 (1989), 141–161. V. Krishnamurthy, On the number of topologies on a finite set, Amer. Math. Monthly 73 (1966), 154–157. G. Liu, M. Kitazawa, M. Eguchi, Y. Fuwa, and Y. Nakamura, Dilation and reduction processing in finite topological spaces and its application to inspection of printed boards, Electronics and Communications in Japan (Part III: Fundamental Electronic Science) 85 (2002), no. 12, 89–100. J. P. May, Simplicial objects in Algebraic Topology, Van Nostrand Mathematical Studies, University of Chicago Press, 1967. J. P. May, Finite spaces and larger contexts (in progress), http://math. uchicago.edu/˜may/REU2020/FINITEBOOK.pdf, 2020. M. Marco-Buzunáriz, Kenzo, https://github.com/ miguelmarco/kenzo, 2015. M. C. McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J. 33 (1966), no. 3, 465–474. J. Milnor, Morse theory. Based on lecture notes by M. Spivak and R. Wells topological spaces, Annals of Mathematics Studies, Princeton University Press, Princeton 51 (1963), 153pp. G. Minian, Some remarks on Morse theory for posets, homological morse theory and finite manifolds, Topology and its Applications 159 (2012), no. 12, 2860–2869. A. Maier, C. Syben, T. Lasser, and C. Riess, A gentle introduction to deep learning in medical image processing, Zeitschrift f¨ur Medizinische Physik 29 (2019), no. 2, 86–101. J. R. Munkres, Elements of Algebraic Topology, AddisonWesley Publishing Company, 1984. OpenAI, Gym, https://gym.openai.com/, 2016. J. Palmieri, Examples of simplicial sets – Sage reference manual v9.2, 2020, https://doc.sagemath.org/html/en/reference/ homology/sage/homology/simplicial_set_examples. html#sage.homology.simplicial_set_examples. simplicial_data_from_kenzo_output. D. Peifer, M. Stillman, and D. Halpern-Leistner, Learning selection strategies in buchberger’s algorithm, https://arxiv.org/pdf/2005. 01917.pdf, 2020. D. Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. Math. 28 (1978), 101–128. J. Renders, Finite topological spaces in algebraic topology, Project of Master of Science in Mathematics, Ghent University, 2019. J. Rotman, An introduction to algebraic topology, Graduate Texts in Mathematics, vol. 119, Springer, 1998. A. Romero, J. Rubio, and F. Sergeraert, Computing spectral sequences, Journal of Symbolic Computation 41 (2006), no. 10, 1059–1079. J. Rubio and F. Sergeraert, Constructive Homological Algebra and Applications, Lecture Notes Summer School on Mathematics, Algorithms, and Proofs, University of Genova, 2006, http://www-fourier.ujf-grenoble.fr/˜sergerar/ Papers/Genova-MAP-2006-v3.pdf. A. Romero and F. Sergeraert, Discrete Vector Fields and Fundamental Algebraic Topology, https://arxiv.org/pdf/1005.5685.pdf, 2010. J. Rubio, F. Sergeraert, and Y. Siret, EAT: Symbolic Software for Effective Homology Computation, Institut Fourier, Grenoble, 1997, https://www-fourier.ujf-grenoble.fr/˜sergerar/ Kenzo/EAT-program.zip. F. Sergeraert, The computability problem in Algebraic Topology, Advances in Mathematics 104 (1994), no. 1, 1–29. M. Shiraki, On finite topological spaces, Rep. Fac. Sci. Kagoshima Univ. 1 (1968), 1–8. D. Silver, A. Huang, and C. Maddison, Mastering the game of Go with deep neural networks and tree search, Nature 529 (2016), 484—-489. L. R. Silverstein, Probability and Machine Learning in Combinatorial Commutative Algebra, Ph.D. thesis, University of California Davis, 2019. R. E. Stong, Finite topological spaces, Trans. Amer. Math. Soc. 123 (1966), no. 2, 325–340. C. J. C. H. Watkins, Learning from Delayed Rewards, Ph.D. thesis, Cambridge University, 1989. J. H. C. Whitehead, Simple homotopy types, Amer. J. Math. 72 (1950), 1–57. G. Whitehead, Fiber spaces and the Eilenberg homology groups, Proceedings of the National Academy of Science of the United States of America 38 (1952), no. 5, 426–430.
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dc.publisher.spa.fl_str_mv	Universidad Nacional de Colombia
dc.publisher.program.spa.fl_str_mv	Bogotá - Ciencias - Doctorado en Ciencias - Matemáticas
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dc.publisher.faculty.spa.fl_str_mv	Facultad de Ciencias
dc.publisher.place.spa.fl_str_mv	Bogotá, Colombia
dc.publisher.branch.spa.fl_str_mv	Universidad Nacional de Colombia - Sede Bogotá
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spelling	Atribución-NoComercial 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Lambán Pardo, Laureanob16d52b3de1cc5e04e2c26fc063d96baRomero Ibáñez, Ana4da831ea0a17b92acacb1960122ea5d9Sarria Zapata, Humbertob42e5cc93a0c826175a443eb6c22e97e600Cuevas Rozo, Julián Leonardo2b2898cfb7e264405deac484054086d46002021-10-13T16:18:33Z2021-10-13T16:18:33Z2021https://repositorio.unal.edu.co/handle/unal/80538Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/ilustraciones, gráficas, tablasHasta el momento, los métodos conocidos para el cálculo de invariantes de espacios topológicos finitos eran aplicables solamente a los posets de caras de complejos simpliciales o de CW-complejos regulares. En este trabajo hemos desarrollado versiones constructivas de algunos resultados teóricos de diferentes autores acerca de espacios finitos, produciendo en particular nuevos algoritmos para el cálculo explícito de algunos complejos de cadenas asociados a espacios finitos h-regulares y sus correspondientes generadores. Hasta donde sabemos, nuestro programa es el único software capaz de calcular grupos de homología de espacios topológicos finitos trabajando directamente sobre los posets. Hemos mejorado nuestros algoritmos sobre espacios finitos h-regulares mediante el uso de campos de vectores discretos, produciendo un nuevo algoritmo para construir dichos campos discretos definidos directamente sobre el poset, además de crear un proceso de h-regularización de espacios finitos permitiendo así ampliar la familia de espacios finitos h-regulares conocidos en la literatura. También hemos presentado una interfaz entre los sistemas de álgebra computacional SageMath y Kenzo. Nuestro trabajo ha permitido que ambos sistemas colaboren mutuamente en algunos cálculos que no pueden ser hechos de manera independiente por dichos programas. Más aún, hemos creado un módulo en SageMath implementando espacios topológicos finitos y algunos conceptos relacionados. Finalmente, hemos considerado algunas estrategias para estudiar diferentes alternativas para calcular campos de vectores discretos de mayor longitud sobre espacios finitos, haciendo uso de algunas técnicas de aprendizaje automático para obtener campos de vectores discretos de la mayor longitud posible. (Texto tomado de la fuente).Up to now, the known methods for computing invariants of finite topological spaces were applicable only for face posets of simplicial complexes or regular CW-complexes. In this work, we have made constructive some theoretical results on finite topological spaces by different authors, producing in particular new algorithms for computing in an explicit way some chain complexes associated with h-regular finite topological spaces and their corresponding generators. Up to our knowledge, our new program is the only software able to compute homology groups of finite topological spaces working directly on the posets. We have improved our algorithms on h-regular spaces by using discrete vector fields, producing a new algorithm for constructing a discrete vector field defined directly on the poset; moreover, we have created a process of h-regularization of finite spaces, allowing to expand the family of h-regular finite spaces known in the literature. We have presented an interface between the computer algebra systems SageMath and Kenzo. Our work has allowed both systems to collaborate in some computations which can not be done independently in any of the programs. Moreover, we have created a module implementing finite topological spaces and related concepts in SageMath. Finally, we have considered some strategies trying to study alternatives to compute longer discrete vector fields on finite spaces, considering some machine learning techniques to obtain discrete vector fields as big as possible.Incluye anexosDoctoradoDoctor en Ciencias - Matemáticasxvi, 137 páginasapplication/pdfengUniversidad Nacional de ColombiaBogotá - Ciencias - Doctorado en Ciencias - MatemáticasDepartamento de MatemáticasFacultad de CienciasBogotá, ColombiaUniversidad Nacional de Colombia - Sede Bogotá510 - Matemáticas::514 - TopologíaTopological spacesInvariantsVector analysisEspacios topológicosInvariantesAnálisis vectorialHomology groupsHomotopy invariantsH-regular finite spacesEffective algorithmsWeak homotopy typesComputational topologyDiscrete vector fieldsEspacios topológicos finitosGrupos de homologíaCampos de vectores discretosInvariantes homotópicosEspacios finitos h-regularesAlgoritmos efectivosTipos de homotopía débilTopología computacionalEffective computation of invariants of finite topological spacesCálculo efectivo de invariantes de espacios topológicos finitosTrabajo de grado - Doctoradoinfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_db06Texthttp://purl.org/redcol/resource_type/TDJ. J. Andrews and M. L. Curtis, Free groups and handlebodies, Proceedings of the American Mathematical Society, American Mathematical Society 16 (1965), no. 2, 192–195.P. Alexandroff, Diskrete R¨aume, Mat. Sb. (N.S.) 2 (1937), 501–518.J. A. Barmak, Algebraic topology of finite topological spaces and applications, Lecture Notes in Mathematics, vol. 2032, Springer, 2011.B. Benedetti and F. H. Lutz, Library of triangulations, http: //page.math.tu-berlin.de/˜lutz/stellar/library_ of_triangulations/, 2017.Y. Bengio, A. Lodi, and A. Prouvost, Machine learning for combinatorial optimization: a methodological tour d’horizon, https://arxiv.org/ pdf/1811.06128.pdf, 2020.N. Cianci and M. Ottina, A new spectral sequence for homology of posets, Topology and its Applications 217 (2017), 1–19.N. Cianci and M. Ottina, Poset splitting and minimality of finite models, Journal of Combinatorial Theory, Series A 157 (2018), 120–161.J. Cuevas-Rozo, Funciones submodulares y matrices en el estudio de los espacios topológicos finitos, Tesis de maestría, Universidad Nacional de Colombia, 2016.J. Cuevas-Rozo, Point reduction algorithms and discrete vector fields for finite topological spaces in the Kenzo system, Proceedings of Fourth EACA International School on Computer Algebra and its Applications, 2018, https://www.usc.es/regaca/eacaschool18/files/ Contributed_talks_EACA2018.pdf, pp. 3–4.J. Cuevas-Rozo, Cálculo de invariantes topológicos sobre espacios topológicos finitos, XXII Congreso Colombiano de Matemáticas 2019, 2019, http://scm.org.co/wp-content/uploads/2019/06/ Cronograma_XXII_CCM2019_V2.pdf.J. Cuevas-Rozo, h-regularización de espacios topológicos finitos, VIII Encuentro de Jóvenes Topólogos, 2019, http://xtsunxet.usc.es/etop2019/ EJTop2019-JCuevas.pdf.J. Cuevas-Rozo, Finite topological spaces in Kenzo, https://github.com/ jcuevas-rozo/finite-topological-spaces, 2020.J. Cuevas-Rozo, J. Divasón, M. Marco-Buzunáriz, and A. Romero, A Kenzo interface for algebraic topology computations in SageMath, ACM Commun. Comput. Algebra 53 (2019), no. 2, 61–64.J. Cuevas-Rozo, J. Divasón, M. Marco-Buzunáriz, and A. Romero, Integration of the Kenzo system within SageMath for new algebraic topology computations, Mathematics, 9(7), 722 (2021).J. Cuevas-Rozo, L. Lambán, A. Romero, and H. Sarria, New algorithms for computing homology of finite topological spaces, Proceedings of 24th Conference on Applications of Computer Algebra - ACA 2018, 2018, pp. 108– 112.J. Cuevas-Rozo, L. Lambán, A. Romero, and H. Sarria, Effective homological computations on finite topological spaces, Applicable Algebra in Engineering, Communication and Computing (2020), In press.J. Cuevas-Rozo, M. Marco-Buzunáriz, and A. Romero, Computing with Kenzo from Sage, Proceedings of the 2019 conference on Effective Methods in Algebraic Geometry, 2019, http://eventos.ucm.es/_files/_event/_12097/ _editorFiles/file/Abstracts_MEGA-alphabetic.pdf, p. 23.The SageMath Developers, The Sage Mathematics Software System (Version 9.2), https://www.sagemath.org, 2020.X. Dousson, J. Rubio, F. Sergeraert, and Y. Siret, The Kenzo program, Institut Fourier, Grenoble, 1999, http://www-fourier. ujf-grenoble.fr/˜sergerar/Kenzo/.X. Fernández, Finite-spaces, https://github.com/ ximenafernandez/Finite-Spaces, 2017.X. Fernández, Métodos combinatorios y algoritmos en topología de dimensiones bajas y la conjetura de Andrews-Curtis, Ph.D. thesis, Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales, 2017.X. Fernández, 3-deformations of 2-complexes and Morse theory, https:// arxiv.org/pdf/1912.00115.pdf, 2019.R. Forman, Morse theory for cell complexes, Advances in Mathematics 134 (1998), 90–145.A. Hatcher, Algebraic topology, Cambridge University Press, 2002.G. Heber, A repackaged version of the Kenzo program by Francis Sergeraert and collaborators, https://github.com/gheber/kenzo, 2019.J. Heras, A Kenzo module computing homotopy groups, 2011, https://esus.unirioja.es/psycotrip/archivos_ documentos/homotopy.cl.M.W. Jahn, P. E. Bradley, M. Al-Doori, and M. Breunig, Topologically consistent models for efficient big geo-spatio-temporal data distribution, ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences IV-4/W5 (2017), 65–72.A. I. Khan and S. Al-Habsi, Machine Learning in Computer Vision, Procedia Computer Science 167 (2020), 1444–1451.V. A. Kovalevsky, Finite topology as applied to image analysis, Computer Vision, Graphics and Image Processing 46 (1989), 141–161.V. Krishnamurthy, On the number of topologies on a finite set, Amer. Math. Monthly 73 (1966), 154–157.G. Liu, M. Kitazawa, M. Eguchi, Y. Fuwa, and Y. Nakamura, Dilation and reduction processing in finite topological spaces and its application to inspection of printed boards, Electronics and Communications in Japan (Part III: Fundamental Electronic Science) 85 (2002), no. 12, 89–100.J. P. May, Simplicial objects in Algebraic Topology, Van Nostrand Mathematical Studies, University of Chicago Press, 1967.J. P. May, Finite spaces and larger contexts (in progress), http://math. uchicago.edu/˜may/REU2020/FINITEBOOK.pdf, 2020.M. Marco-Buzunáriz, Kenzo, https://github.com/ miguelmarco/kenzo, 2015.M. C. McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J. 33 (1966), no. 3, 465–474.J. Milnor, Morse theory. Based on lecture notes by M. Spivak and R. Wells topological spaces, Annals of Mathematics Studies, Princeton University Press, Princeton 51 (1963), 153pp.G. Minian, Some remarks on Morse theory for posets, homological morse theory and finite manifolds, Topology and its Applications 159 (2012), no. 12, 2860–2869.A. Maier, C. Syben, T. Lasser, and C. Riess, A gentle introduction to deep learning in medical image processing, Zeitschrift f¨ur Medizinische Physik 29 (2019), no. 2, 86–101.J. R. Munkres, Elements of Algebraic Topology, AddisonWesley Publishing Company, 1984.OpenAI, Gym, https://gym.openai.com/, 2016.J. Palmieri, Examples of simplicial sets – Sage reference manual v9.2, 2020, https://doc.sagemath.org/html/en/reference/ homology/sage/homology/simplicial_set_examples. html#sage.homology.simplicial_set_examples. simplicial_data_from_kenzo_output.D. Peifer, M. Stillman, and D. Halpern-Leistner, Learning selection strategies in buchberger’s algorithm, https://arxiv.org/pdf/2005. 01917.pdf, 2020.D. Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. Math. 28 (1978), 101–128.J. 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Whitehead, Fiber spaces and the Eilenberg homology groups, Proceedings of the National Academy of Science of the United States of America 38 (1952), no. 5, 426–430.InvestigadoresPúblico generalLICENSElicense.txtlicense.txttext/plain; charset=utf-83964https://repositorio.unal.edu.co/bitstream/unal/80538/1/license.txtcccfe52f796b7c63423298c2d3365fc6MD51ORIGINAL1018454513.2021.pdf1018454513.2021.pdfTesis de Doctorado en Ciencias - Matemáticasapplication/pdf1259340https://repositorio.unal.edu.co/bitstream/unal/80538/2/1018454513.2021.pdf6b9efe9330848387fd3f9d96f860fddeMD52THUMBNAIL1018454513.2021.pdf.jpg1018454513.2021.pdf.jpgGenerated Thumbnailimage/jpeg4225https://repositorio.unal.edu.co/bitstream/unal/80538/3/1018454513.2021.pdf.jpg2d792f67e922fe22e752d12d370d8af2MD53unal/80538oai:repositorio.unal.edu.co:unal/805382023-07-30 23:03:40.92Repositorio Institucional Universidad Nacional de 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Effective computation of invariants of finite topological spaces

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