Schottky Problem

Ilustraciones

Autores:: Echavarría Arenas, Santiago

Tipo de recurso:

Fecha de publicación:: 2023

Institución:: Universidad Nacional de Colombia

Repositorio:: Universidad Nacional de Colombia

Idioma:: eng

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dc.title.eng.fl_str_mv	Schottky Problem
dc.title.translated.spa.fl_str_mv	Problema de Schottky
title	Schottky Problem
spellingShingle	Schottky Problem 510 - Matemáticas::515 - Análisis Variedades (Matemáticas) Variedades diferenciales Variedades de Riemann Funciones theta Funciones holomorfas Funciones abelianas Funciones de variable compleja Abelian variety Jacobian variety Theta functions Fay's trisecant identity Variedad abeliana Variedad Jacobiana funciones theta Identidad trisecante de Fay
title_short	Schottky Problem
title_full	Schottky Problem
title_fullStr	Schottky Problem
title_full_unstemmed	Schottky Problem
title_sort	Schottky Problem
dc.creator.fl_str_mv	Echavarría Arenas, Santiago
dc.contributor.advisor.none.fl_str_mv	Quintero Vélez, Alexander
dc.contributor.author.none.fl_str_mv	Echavarría Arenas, Santiago
dc.subject.ddc.spa.fl_str_mv	510 - Matemáticas::515 - Análisis
topic	510 - Matemáticas::515 - Análisis Variedades (Matemáticas) Variedades diferenciales Variedades de Riemann Funciones theta Funciones holomorfas Funciones abelianas Funciones de variable compleja Abelian variety Jacobian variety Theta functions Fay's trisecant identity Variedad abeliana Variedad Jacobiana funciones theta Identidad trisecante de Fay
dc.subject.lemb.none.fl_str_mv	Variedades (Matemáticas) Variedades diferenciales Variedades de Riemann Funciones theta Funciones holomorfas Funciones abelianas Funciones de variable compleja
dc.subject.proposal.eng.fl_str_mv	Abelian variety Jacobian variety Theta functions Fay's trisecant identity
dc.subject.proposal.spa.fl_str_mv	Variedad abeliana Variedad Jacobiana funciones theta Identidad trisecante de Fay
description	Ilustraciones
publishDate	2023
dc.date.issued.none.fl_str_mv	2023-08
dc.date.accessioned.none.fl_str_mv	2024-10-21T12:48:29Z
dc.date.available.none.fl_str_mv	2024-10-21T12:48:29Z
dc.type.spa.fl_str_mv	Trabajo de grado - Maestría
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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.indexed.spa.fl_str_mv	LaReferencia
dc.relation.references.spa.fl_str_mv	Enrico Arbarello and Corrado De Concini. “On a set of equations characterizing Riemann matrices”. In: Annals of Mathematics (1984), pp. 119–140. Enrico Arbarello and Corrado De Concini. “Geometrical aspects of the KP equation”. In: Global Geometry and Mathematical Physics: Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (CIME) held at Montecatini Terme, Italy, July 4–12, 1988. Springer. 1990, pp. 95– 137. Arnaud Beauville. “Theta functions, old and new”. In: HAL 2013 (2013). Christina Birkenhake and Herbert Lange. Complex abelian varieties. Vol. 6. Springer, 2004. Jean-Pierre Demailly. Complex analytic and differential geometry. Citeseer, 1997. Boris Anatol’evich Dubrovin. “Theta functions and non-linear equations”. In: Russian mathematical surveys 36.2 (1981), p. 11. John D Fay. Theta functions on Riemann surfaces. Vol. 352. Springer, 2006. Hershel M Farkas and Irwin Kra. Riemann surfaces. Springer, 1992. Phillip Griffiths and Joseph Harris. Principles of algebraic geometry. John Wiley & Sons, 2014. Robert Clifford Gunning and Hugo Rossi. Analytic functions of several complex variables. Vol. 368. American Mathematical Soc., 2009. Hans Grauert and Reinhold Remmert. Coherent analytic sheaves. Vol. 265. Springer Science & Business Media, 2012. Robert Clifford Gunning. “Some curves in abelian varities”. In: Inventiones mathematicae 66 (1982), pp. 377–389. Daniel Huybrechts. Complex geometry: an introduction. Vol. 78. Springer, 2005. Jurgen Jost. Compact riemann surfaces. Springer, 2006. Igor Moiseevich Krichever. “Integration of nonlinear equations by the methods of algebraic geometry”. In: Funktsional’nyi Analiz i ego Prilozheniya 11.1 (1977), pp. 15–31. E. Keith Lloyd. Encyclopedia of Mathematics. Bell Polynomial. url: https: //encyclopediaofmath.org/wiki/Bell_polynomial. David Mumford, John Fogarty, and Frances Kirwan. Geometric invariant theory. Vol. 34. Springer Science & Business Media, 1994. David Mumford and C Musili. Tata lectures on theta. I (Modern Birkh¨auser classics). Birkh¨auser Boston Incorporated, 2007. David Mumford, Chidambaran Padmanabhan Ramanujam, and Jurij Ivanoviˇc Manin. Abelian varieties. Vol. 5. Oxford university press Oxford, 1974. Motohico Mulase. “Cohomological structure in soliton equations and Jacobian varieties”. In: Journal of Differential Geometry 19.2 (1984), pp. 403– 430. David Mumford et al. Tata Lectures on Theta II: Jacobian theta functions and differential equations, Modern Birkh¨auser Classics. 2012. Cris Poor. “Fay’s trisecant formula and cross-ratios”. In: Proceedings of the American Mathematical Society 114.3 (1992), pp. 667–671. Ryuji Sasaki. “Modular forms vanishing at the reducible points of the Siegel upper-half space.” In: (1983). Takahiro Shiota. “Characterization of Jacobian varieties in terms of soliton equations”. In: Inventiones mathematicae 83.2 (1986), pp. 333–382. Herbert Seifert and William Threlfall. Seifert and threlfal: A textbook of Topolog and Seifert: topology of 3-Dimensional Fibered Spaces. 1980. Gerald E Welters. “On flexes of the Kummer variety (note on a theorem of RC Gunning)”. In: Indagationes Mathematicae (Proceedings). Vol. 86. 4. Elsevier. 1983, pp. 501–520. Gerald E Welters. “A criterion for Jacobi varieties”. In: Annals of Mathematics 120.3 (1984), pp. 497–504. Raymond O’Neil Wells and Oscar Garc´ıa-Prada. Differential analysis on complex manifolds. Vol. 21980. Springer New York, 1980.
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dc.format.extent.spa.fl_str_mv	74 páginas
dc.format.mimetype.spa.fl_str_mv	application/pdf
dc.publisher.spa.fl_str_mv	Universidad Nacional de Colombia
dc.publisher.program.spa.fl_str_mv	Medellín - Ciencias - Maestría en Ciencias - Matemáticas
dc.publisher.faculty.spa.fl_str_mv	Facultad de Ciencias
dc.publisher.place.spa.fl_str_mv	Medellín, Colombia
dc.publisher.branch.spa.fl_str_mv	Universidad Nacional de Colombia - Sede Medellín
institution	Universidad Nacional de Colombia
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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_abf2Quintero Vélez, Alexanderef7b285211024ca34a4305c09822e9bbEchavarría Arenas, Santiago43d82a3864cb727fb95c316e84d628342024-10-21T12:48:29Z2024-10-21T12:48:29Z2023-08https://repositorio.unal.edu.co/handle/unal/87014Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/IlustracionesAn accessible introduction to the Schottky problem is given, with explicit computations included. The Schottky problem asks what abelian varieties are jacobian varieties, where a jacobian variety is a certain complex torus constructed out from a Riemann surface, and abelian varieties are complex tori which can be embedded in projective space. Ideas around embeddings will be introduced. Fay’s trisecant identity, which is an identity that comes from generalizing the cross ratio of a Riemann sphere to higher genus via theta functions, will be the cornerstone pointing towards the statements that solved the Schottky problem in a concrete way in the 1980’s. (Tomado de la fuente)Se dará una introducción accesible al problema de Schottky con cómputos explícitos. El problema de Schottky consiste en ver que variedades abelianas son variedades jacobianas, en donde una variedad jacobiana es un cierto toro complejo construido a través de una superficie de Riemann, y las variedades abelianas son toros complejos que se pueden embeber en el espacio proyectivo. Las ideas alrededor de los embebimientos serán introducidas. Explorar la identidad trisecante de Fay, que viene de una generalización de la razón cruzada de la esfera de Riemann a superficies de Riemann de géneros mayores a través de funciones theta, será la base para introducir los teoremas que solucionaron el problema de Schottky de forma concreta en la decada de 1980.MaestríaMagister en Ciencias-MatemáticasMatemáticas.Sede Medellín74 páginasapplication/pdfengUniversidad Nacional de ColombiaMedellín - Ciencias - Maestría en Ciencias - MatemáticasFacultad de CienciasMedellín, ColombiaUniversidad Nacional de Colombia - Sede Medellín510 - Matemáticas::515 - AnálisisVariedades (Matemáticas)Variedades diferencialesVariedades de RiemannFunciones thetaFunciones holomorfasFunciones abelianasFunciones de variable complejaAbelian varietyJacobian varietyTheta functionsFay's trisecant identityVariedad abelianaVariedad Jacobianafunciones thetaIdentidad trisecante de FaySchottky ProblemProblema de SchottkyTrabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TMLaReferenciaEnrico Arbarello and Corrado De Concini. “On a set of equations characterizing Riemann matrices”. In: Annals of Mathematics (1984), pp. 119–140.Enrico Arbarello and Corrado De Concini. “Geometrical aspects of the KP equation”. In: Global Geometry and Mathematical Physics: Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (CIME) held at Montecatini Terme, Italy, July 4–12, 1988. Springer. 1990, pp. 95– 137.Arnaud Beauville. “Theta functions, old and new”. In: HAL 2013 (2013).Christina Birkenhake and Herbert Lange. Complex abelian varieties. Vol. 6. Springer, 2004.Jean-Pierre Demailly. Complex analytic and differential geometry. Citeseer, 1997.Boris Anatol’evich Dubrovin. “Theta functions and non-linear equations”. In: Russian mathematical surveys 36.2 (1981), p. 11.John D Fay. Theta functions on Riemann surfaces. Vol. 352. Springer, 2006.Hershel M Farkas and Irwin Kra. Riemann surfaces. Springer, 1992.Phillip Griffiths and Joseph Harris. Principles of algebraic geometry. John Wiley & Sons, 2014.Robert Clifford Gunning and Hugo Rossi. Analytic functions of several complex variables. Vol. 368. American Mathematical Soc., 2009.Hans Grauert and Reinhold Remmert. Coherent analytic sheaves. Vol. 265. Springer Science & Business Media, 2012.Robert Clifford Gunning. “Some curves in abelian varities”. In: Inventiones mathematicae 66 (1982), pp. 377–389.Daniel Huybrechts. Complex geometry: an introduction. Vol. 78. Springer, 2005.Jurgen Jost. Compact riemann surfaces. Springer, 2006.Igor Moiseevich Krichever. “Integration of nonlinear equations by the methods of algebraic geometry”. In: Funktsional’nyi Analiz i ego Prilozheniya 11.1 (1977), pp. 15–31.E. Keith Lloyd. Encyclopedia of Mathematics. Bell Polynomial. url: https: //encyclopediaofmath.org/wiki/Bell_polynomial.David Mumford, John Fogarty, and Frances Kirwan. Geometric invariant theory. Vol. 34. Springer Science & Business Media, 1994.David Mumford and C Musili. Tata lectures on theta. I (Modern Birkh¨auser classics). Birkh¨auser Boston Incorporated, 2007.David Mumford, Chidambaran Padmanabhan Ramanujam, and Jurij Ivanoviˇc Manin. Abelian varieties. Vol. 5. Oxford university press Oxford, 1974.Motohico Mulase. “Cohomological structure in soliton equations and Jacobian varieties”. In: Journal of Differential Geometry 19.2 (1984), pp. 403– 430.David Mumford et al. Tata Lectures on Theta II: Jacobian theta functions and differential equations, Modern Birkh¨auser Classics. 2012.Cris Poor. “Fay’s trisecant formula and cross-ratios”. In: Proceedings of the American Mathematical Society 114.3 (1992), pp. 667–671.Ryuji Sasaki. “Modular forms vanishing at the reducible points of the Siegel upper-half space.” In: (1983).Takahiro Shiota. “Characterization of Jacobian varieties in terms of soliton equations”. In: Inventiones mathematicae 83.2 (1986), pp. 333–382.Herbert Seifert and William Threlfall. Seifert and threlfal: A textbook of Topolog and Seifert: topology of 3-Dimensional Fibered Spaces. 1980.Gerald E Welters. “On flexes of the Kummer variety (note on a theorem of RC Gunning)”. In: Indagationes Mathematicae (Proceedings). Vol. 86. 4. Elsevier. 1983, pp. 501–520.Gerald E Welters. “A criterion for Jacobi varieties”. In: Annals of Mathematics 120.3 (1984), pp. 497–504.Raymond O’Neil Wells and Oscar Garc´ıa-Prada. Differential analysis on complex manifolds. Vol. 21980. Springer New York, 1980.EstudiantesInvestigadoresMaestrosLICENSElicense.txtlicense.txttext/plain; charset=utf-85879https://repositorio.unal.edu.co/bitstream/unal/87014/1/license.txteb34b1cf90b7e1103fc9dfd26be24b4aMD51ORIGINAL1017247829.2024.pdf1017247829.2024.pdfTesis de Maestría en Ciencias - Matemáticasapplication/pdf838856https://repositorio.unal.edu.co/bitstream/unal/87014/2/1017247829.2024.pdf79619d2d7115d931aef11933537c47abMD52THUMBNAIL1017247829.2024.pdf.jpg1017247829.2024.pdf.jpgGenerated Thumbnailimage/jpeg5120https://repositorio.unal.edu.co/bitstream/unal/87014/3/1017247829.2024.pdf.jpgf9ae710b52dcbf6004cee7fc0e66c6f4MD53unal/87014oai:repositorio.unal.edu.co:unal/870142024-10-22 00:11:53.22Repositorio Institucional Universidad Nacional de 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