Categorification of Chern-Weil theory and equivariant cohomology
diagramas
- Autores:
-
Pineda Montoya, Santiago
- Tipo de recurso:
- Doctoral thesis
- Fecha de publicación:
- 2022
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/83348
- Palabra clave:
- 510 - Matemáticas
510 - Matemáticas::516 - Geometría
Geometría diferencial
Geometry, differential
Sistemas locales
Álgebra homotópica
Homomorfismo de Chern-Weil
Cohomología equivariante
- Rights
- openAccess
- License
- Atribución-NoComercial-SinDerivadas 4.0 Internacional
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Categorification of Chern-Weil theory and equivariant cohomology |
| dc.title.translated.spa.fl_str_mv |
Categorificación de la teoría de Chern-Weil y la cohomología equivariante |
| title |
Categorification of Chern-Weil theory and equivariant cohomology |
| spellingShingle |
Categorification of Chern-Weil theory and equivariant cohomology 510 - Matemáticas 510 - Matemáticas::516 - Geometría Geometría diferencial Geometry, differential Sistemas locales Álgebra homotópica Homomorfismo de Chern-Weil Cohomología equivariante |
| title_short |
Categorification of Chern-Weil theory and equivariant cohomology |
| title_full |
Categorification of Chern-Weil theory and equivariant cohomology |
| title_fullStr |
Categorification of Chern-Weil theory and equivariant cohomology |
| title_full_unstemmed |
Categorification of Chern-Weil theory and equivariant cohomology |
| title_sort |
Categorification of Chern-Weil theory and equivariant cohomology |
| dc.creator.fl_str_mv |
Pineda Montoya, Santiago |
| dc.contributor.advisor.none.fl_str_mv |
Quintero Vélez, Alexander Arias Abad, Camilo |
| dc.contributor.author.none.fl_str_mv |
Pineda Montoya, Santiago |
| dc.contributor.orcid.spa.fl_str_mv |
Arias Abad, Camilo [0000-0003-3624-9396] |
| dc.subject.ddc.spa.fl_str_mv |
510 - Matemáticas 510 - Matemáticas::516 - Geometría |
| topic |
510 - Matemáticas 510 - Matemáticas::516 - Geometría Geometría diferencial Geometry, differential Sistemas locales Álgebra homotópica Homomorfismo de Chern-Weil Cohomología equivariante |
| dc.subject.lemb.none.fl_str_mv |
Geometría diferencial Geometry, differential |
| dc.subject.proposal.spa.fl_str_mv |
Sistemas locales Álgebra homotópica Homomorfismo de Chern-Weil Cohomología equivariante |
| description |
diagramas |
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2022 |
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2022-06-28 |
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2023-02-07T14:02:58Z |
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2023-02-07T14:02:58Z |
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Trabajo de grado - Doctorado |
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Universidad Nacional de Colombia |
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eng |
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eng |
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RedCol LaReferencia |
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M. Sugawara. On a condition that a space is an H-space, Math. J. Okayama Univ., 6:109–129, 1957 J. Stasheff. Homotopy associativity of h-spaces. i, Transactions of the American Mathematical Society 108 (01 1963), 275–292 J. Stasheff. Homotopy associativity of h-spaces. ii, Transactions of the American Mathematical Society 108 (08 1963), 275 J. Stasheff. Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras, Quantum groups Number 1510 in Lecture Notes in Math. Springer, Berlin, 1992 T. Lada, J. Stasheff. Introduction to sh Lie algebras for physicists, Int. J. Theo. Phys. 32 (1993), 1087–1103 B. Zwiebach. Closed string field theory: Quantum action and the B-V master equation, Nucl.Phys. B390 (1993) 33 Holstein, Julian V. Morita cohomology, Cambridge University Press, 2015 Jonathan Block and Aaron M. Smith. The higher Riemann-Hilbert correspondence, Adv. Math., 2014 Abad, Camilo Arias and Schatz, Florian. The ¨ A• de Rham theorem and integration of representations up to homotopy, Int. Math. Res. Notices, 2013 Abad, Camilo Arias and Schatz, Florian. Flat Z-graded connections and loop spaces, International Mathematics ¨ Research Notices, 2018 C. Arias Abad, A. Quintero Velez and S. V ´ elez V ´ asquez. An ´ A•-version of the Poincare lemma. Pacific Journal ´ of Mathematics, 2019 S.S. Chern. Differential geometry of fiber bundles. Proceedings of the International Congress of Mathematicians, Cambridge, Mass., vol. 2, pages 397-411, Amer. Math. Soc., Providence, R. I. 1950 A. Weil. Geom ´ etrie diff ´ erentielle des espaces fibres. unpublished, 1949 H. Cartan. La transgression dans un groupe de Lie et dans un fibre principal. Colloque de topologie (espaces ´ fibres) (Bruxelles), Centre belge de recherches math ´ ematiques, Georges Thone, Li ´ ege, Masson et Cie., Paris, ` 1950 H. Cartan. Notions d’algebre diff ` erentiel le; application aux groupes de Lie et aux vari ´ et ´ es o ´ u op ` ere un groupe de ` Lie. Colloque de topologie (espaces fibres) (Bruxelles), Georges Thone, Li ´ ege, Masson et Cie., Paris, 1950. A. Borel. Seminar on transformation groups. Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960 C. Arias Abad, S. Pineda Montoya and A. Quintero Velez. Chern-Weil theory for ´ •-local systems. arXiv:2105.00461, submitted for publication C. Arias Abad, S. Pineda Montoya and A. Quintero Velez. Equivariant de Rham Theorem for ´ •-local systems. In preparation C. Arias Abad, A. Quintero Velez. Singular chains on Lie groups and the Cartan relations II. preprint C. Arias Abad. Singular chains on Lie groups and the Cartan relations I. arXiv:1908.10460, submitted for publication Eckhard Meinrenken. Clifford algebras and Lie theory, Springer, 2013 Guillemin, Victor W and Sternberg, Shlomo. Supersymmetry and equivariant de Rham theory, Springer Science & Business Media, 2013 Reinhold, Ben. L-•-algebras and their cohomology, Emergent Scientist, 2019 Keller, Bernhard. On differential graded categories, International Congress of Mathematicians. Vol. II, 2006 J. Block, A. Smith. The Riemann-Hilbert correspondence for infinity local systems, Advances in Mathematics, 2009 Arias Abad, Camilo and Crainic, Marius. Representations up to homotopy of Lie algebroids, J. Reine Angew. Math. (Crelle’s Journal), 2012 P. Seidel. Fukaya categories and Picard Lefschetz theory, Zurich Lectures in Advanced Mathematics, EMS, 2008 Mehta, Rajan Amit and Zambon, Marco. L•-algebra actions, Differ. Geom. Appl., 2012 Loring W. TU. Introductory Lectures on Equivariant Cohomology, Princeton University Press, 2020 MathOverflow. Why does the singular simplicial space geometrically realize to the original space?. https://mathoverflow.net/questions/171662/why-does-the-singular-simplicial-space-geometrically-realize-tothe-original-spa Goerss, Paul G. and Jardine, John F. Simplicial homotopy theory, Springer Science & Business Media, 2009 Joyal, Andre and Tierney, Myles. Notes on simplicial homotopy theory, ´ http://mat.uab.cat/ kock/crm/hocat/advanced-course/Quadern47.pdf, 2008 Ruschoff, Christian. Notes on simplicial homotopy theory, https://www.mathi.uni- ¨ heidelberg.de/ rueschoff/ss17sset/sset.pdf, 2017 Jardine, John Frederick. Simplicial presheaves, Journal of Pure and Applied Algebra, 1987 Hatcher, Allen. Algebraic topology, Cambridge University Press, 2005 |
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92 páginas |
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Universidad Nacional de Colombia |
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Medellín - Ciencias - Doctorado en Ciencias - Matemáticas |
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Facultad de Ciencias |
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Medellín, Colombia |
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Universidad Nacional de Colombia - Sede Medellín |
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Atribución-NoComercial-SinDerivadas 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Quintero Vélez, Alexanderef7b285211024ca34a4305c09822e9bbArias Abad, Camilo13850311a020fe0b3dfcaf92c87c406ePineda Montoya, Santiagobed72974b97883bae9501e93ca1d1cd2Arias Abad, Camilo [0000-0003-3624-9396]2023-02-07T14:02:58Z2023-02-07T14:02:58Z2022-06-28https://repositorio.unal.edu.co/handle/unal/83348Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/diagramasEsta tesis contempla la generalización de resultados de geomtría diferencial clásica en el contexto de los sistemas locales homotópicos. En particular, se realiza la construcción del homomorfismo de Chern-Weil y el teorema equivariante de de Rham en el contexto de las categorias diferenciales graduadas conformadas por los sistemas locales homotópicos. (Texto tomado de la fuente)Let G be a compact connected Lie group acting on a smooth manifold M. We show that the DG categories Loc∞(BG) and Loc∞(MG) of ∞-local systems on the classifying space of G and the homotopy quotient of M, respectively, can be described infinitesimally as the categories InfLoc∞(g) of basic g-L∞ spaces and InfLoc∞(g,M) of g graded G-equivariant vector bundles, respectively. Moreover, we show that, given a principal bundle π : P → X with structure group G and any connection θ on P, there are DG functors C Wθ : InfLoc∞(g) −→ Loc∞(X), and Cθ : InfLoc∞(g,M) −→ Loc∞((P× M)/G), that corresponds to the pullback functor by the classifying map of P. An A∞-natural isomorphism relates the functors associated with different connections. This construction categorizes the ChernWeil homomorphism, which is recovered by applying the functor C Wθ to the endomorphisms of the constant local system. Finally, we obtain a categorification of the equivariant de Rham theorem for infinity local systems, namely, the A∞-fuctor DR : InfLoc∞(g,M) → Loc∞(MG).DoctoradoDoctor en Ciencias - MatemáticasÁrea Curricular en Matemáticas92 páginasapplication/pdfengUniversidad Nacional de ColombiaMedellín - Ciencias - Doctorado en Ciencias - MatemáticasFacultad de CienciasMedellín, ColombiaUniversidad Nacional de Colombia - Sede Medellín510 - Matemáticas510 - Matemáticas::516 - GeometríaGeometría diferencialGeometry, differentialSistemas localesÁlgebra homotópicaHomomorfismo de Chern-WeilCohomología equivarianteCategorification of Chern-Weil theory and equivariant cohomologyCategorificación de la teoría de Chern-Weil y la cohomología equivarianteTrabajo de grado - Doctoradoinfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_db06Texthttp://purl.org/redcol/resource_type/TDRedColLaReferenciaM. Sugawara. On a condition that a space is an H-space, Math. J. Okayama Univ., 6:109–129, 1957J. Stasheff. Homotopy associativity of h-spaces. i, Transactions of the American Mathematical Society 108 (01 1963), 275–292J. Stasheff. Homotopy associativity of h-spaces. ii, Transactions of the American Mathematical Society 108 (08 1963), 275J. Stasheff. Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras, Quantum groups Number 1510 in Lecture Notes in Math. Springer, Berlin, 1992T. Lada, J. Stasheff. Introduction to sh Lie algebras for physicists, Int. J. Theo. Phys. 32 (1993), 1087–1103B. Zwiebach. Closed string field theory: Quantum action and the B-V master equation, Nucl.Phys. B390 (1993) 33Holstein, Julian V. Morita cohomology, Cambridge University Press, 2015Jonathan Block and Aaron M. Smith. The higher Riemann-Hilbert correspondence, Adv. Math., 2014Abad, Camilo Arias and Schatz, Florian. The ¨ A• de Rham theorem and integration of representations up to homotopy, Int. Math. Res. Notices, 2013Abad, Camilo Arias and Schatz, Florian. Flat Z-graded connections and loop spaces, International Mathematics ¨ Research Notices, 2018C. Arias Abad, A. Quintero Velez and S. V ´ elez V ´ asquez. An ´ A•-version of the Poincare lemma. Pacific Journal ´ of Mathematics, 2019S.S. Chern. Differential geometry of fiber bundles. Proceedings of the International Congress of Mathematicians, Cambridge, Mass., vol. 2, pages 397-411, Amer. Math. Soc., Providence, R. I. 1950A. Weil. Geom ´ etrie diff ´ erentielle des espaces fibres. unpublished, 1949H. Cartan. La transgression dans un groupe de Lie et dans un fibre principal. Colloque de topologie (espaces ´ fibres) (Bruxelles), Centre belge de recherches math ´ ematiques, Georges Thone, Li ´ ege, Masson et Cie., Paris, ` 1950H. Cartan. Notions d’algebre diff ` erentiel le; application aux groupes de Lie et aux vari ´ et ´ es o ´ u op ` ere un groupe de ` Lie. Colloque de topologie (espaces fibres) (Bruxelles), Georges Thone, Li ´ ege, Masson et Cie., Paris, 1950.A. Borel. Seminar on transformation groups. Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960C. Arias Abad, S. Pineda Montoya and A. Quintero Velez. Chern-Weil theory for ´ •-local systems. arXiv:2105.00461, submitted for publicationC. Arias Abad, S. Pineda Montoya and A. Quintero Velez. Equivariant de Rham Theorem for ´ •-local systems. In preparationC. Arias Abad, A. Quintero Velez. Singular chains on Lie groups and the Cartan relations II. preprintC. Arias Abad. Singular chains on Lie groups and the Cartan relations I. arXiv:1908.10460, submitted for publicationEckhard Meinrenken. Clifford algebras and Lie theory, Springer, 2013Guillemin, Victor W and Sternberg, Shlomo. Supersymmetry and equivariant de Rham theory, Springer Science & Business Media, 2013Reinhold, Ben. L-•-algebras and their cohomology, Emergent Scientist, 2019Keller, Bernhard. On differential graded categories, International Congress of Mathematicians. Vol. II, 2006J. Block, A. Smith. The Riemann-Hilbert correspondence for infinity local systems, Advances in Mathematics, 2009Arias Abad, Camilo and Crainic, Marius. Representations up to homotopy of Lie algebroids, J. Reine Angew. Math. (Crelle’s Journal), 2012P. Seidel. Fukaya categories and Picard Lefschetz theory, Zurich Lectures in Advanced Mathematics, EMS, 2008Mehta, Rajan Amit and Zambon, Marco. L•-algebra actions, Differ. Geom. Appl., 2012Loring W. TU. Introductory Lectures on Equivariant Cohomology, Princeton University Press, 2020MathOverflow. Why does the singular simplicial space geometrically realize to the original space?. https://mathoverflow.net/questions/171662/why-does-the-singular-simplicial-space-geometrically-realize-tothe-original-spaGoerss, Paul G. and Jardine, John F. Simplicial homotopy theory, Springer Science & Business Media, 2009Joyal, Andre and Tierney, Myles. Notes on simplicial homotopy theory, ´ http://mat.uab.cat/ kock/crm/hocat/advanced-course/Quadern47.pdf, 2008Ruschoff, Christian. Notes on simplicial homotopy theory, https://www.mathi.uni- ¨ heidelberg.de/ rueschoff/ss17sset/sset.pdf, 2017Jardine, John Frederick. Simplicial presheaves, Journal of Pure and Applied Algebra, 1987Hatcher, Allen. Algebraic topology, Cambridge University Press, 2005ColfuturoInvestigadoresORIGINAL1035425041.2022.pdf1035425041.2022.pdfTesis de Doctorado en matemáticasapplication/pdf704347https://repositorio.unal.edu.co/bitstream/unal/83348/6/1035425041.2022.pdfa69ac1c198c43fc428b1305a90d8b6c4MD56LICENSElicense.txtlicense.txttext/plain; charset=utf-85879https://repositorio.unal.edu.co/bitstream/unal/83348/5/license.txteb34b1cf90b7e1103fc9dfd26be24b4aMD55THUMBNAIL1035425041.2022.pdf.jpg1035425041.2022.pdf.jpgGenerated Thumbnailimage/jpeg4311https://repositorio.unal.edu.co/bitstream/unal/83348/7/1035425041.2022.pdf.jpg09f2c645131fd977e27044350458fbb2MD57unal/83348oai:repositorio.unal.edu.co:unal/833482023-08-15 23:04:39.105Repositorio Institucional Universidad Nacional de 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