Opers as a generalisation of complex projective structures
Essentially, complex projective structures arise as geometries modeled in the projective line. Meanwhile, opers appear as local systems of representation-theoretic nature and are linked to principal bundles with connections. These objects where first introduced in the punctured disk by Drinfeld and...
- Autores:
-
Aragón Rodríguez, Manuel Alejandro
- Tipo de recurso:
- Trabajo de grado de pregrado
- Fecha de publicación:
- 2024
- Institución:
- Universidad de los Andes
- Repositorio:
- Séneca: repositorio Uniandes
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.uniandes.edu.co:1992/73825
- Acceso en línea:
- https://hdl.handle.net/1992/73825
- Palabra clave:
- Opers
Principal bundles
Complex geometry
Lie groups
Riemann surfaces
Matemáticas
- Rights
- openAccess
- License
- Attribution-NonCommercial-NoDerivatives 4.0 International
| id |
UNIANDES2_6c8423385c55271f5cb0d4d20e3e5a56 |
|---|---|
| oai_identifier_str |
oai:repositorio.uniandes.edu.co:1992/73825 |
| network_acronym_str |
UNIANDES2 |
| network_name_str |
Séneca: repositorio Uniandes |
| repository_id_str |
|
| dc.title.eng.fl_str_mv |
Opers as a generalisation of complex projective structures |
| title |
Opers as a generalisation of complex projective structures |
| spellingShingle |
Opers as a generalisation of complex projective structures Opers Principal bundles Complex geometry Lie groups Riemann surfaces Matemáticas |
| title_short |
Opers as a generalisation of complex projective structures |
| title_full |
Opers as a generalisation of complex projective structures |
| title_fullStr |
Opers as a generalisation of complex projective structures |
| title_full_unstemmed |
Opers as a generalisation of complex projective structures |
| title_sort |
Opers as a generalisation of complex projective structures |
| dc.creator.fl_str_mv |
Aragón Rodríguez, Manuel Alejandro |
| dc.contributor.advisor.none.fl_str_mv |
Schaffhauser, Florent Marie Roland Cardona Guio, Alexander |
| dc.contributor.author.none.fl_str_mv |
Aragón Rodríguez, Manuel Alejandro |
| dc.contributor.jury.none.fl_str_mv |
Malakhaltsev, Mikhail |
| dc.subject.keyword.eng.fl_str_mv |
Opers Principal bundles Complex geometry Lie groups Riemann surfaces |
| topic |
Opers Principal bundles Complex geometry Lie groups Riemann surfaces Matemáticas |
| dc.subject.themes.spa.fl_str_mv |
Matemáticas |
| description |
Essentially, complex projective structures arise as geometries modeled in the projective line. Meanwhile, opers appear as local systems of representation-theoretic nature and are linked to principal bundles with connections. These objects where first introduced in the punctured disk by Drinfeld and Sokolov in their study of KdV-type hierarchies and were later given a coordinate-free description for a general reductive group by Beilinson and Drinfeld. Recently, they have been studied in their connection to the geometrical Langlands correspondence and their role in the theory of vertex algebras, as developed by Frenkel. In this thesis, we study the connection between complex projective structures and opers in a geometric and algebraic way and explain how opers generalise complex projective structures when a general complex Lie group is studied. |
| publishDate |
2024 |
| dc.date.accessioned.none.fl_str_mv |
2024-02-02T18:20:51Z |
| dc.date.available.none.fl_str_mv |
2024-02-02T18:20:51Z |
| dc.date.issued.none.fl_str_mv |
2024-01-17 |
| dc.type.none.fl_str_mv |
Trabajo de grado - Pregrado |
| dc.type.driver.none.fl_str_mv |
info:eu-repo/semantics/bachelorThesis |
| dc.type.version.none.fl_str_mv |
info:eu-repo/semantics/acceptedVersion |
| dc.type.coar.none.fl_str_mv |
http://purl.org/coar/resource_type/c_7a1f |
| dc.type.content.none.fl_str_mv |
Text |
| dc.type.redcol.none.fl_str_mv |
http://purl.org/redcol/resource_type/TP |
| format |
http://purl.org/coar/resource_type/c_7a1f |
| status_str |
acceptedVersion |
| dc.identifier.uri.none.fl_str_mv |
https://hdl.handle.net/1992/73825 |
| dc.identifier.instname.none.fl_str_mv |
instname:Universidad de los Andes |
| dc.identifier.reponame.none.fl_str_mv |
reponame:Repositorio Institucional Séneca |
| dc.identifier.repourl.none.fl_str_mv |
repourl:https://repositorio.uniandes.edu.co/ |
| url |
https://hdl.handle.net/1992/73825 |
| identifier_str_mv |
instname:Universidad de los Andes reponame:Repositorio Institucional Séneca repourl:https://repositorio.uniandes.edu.co/ |
| dc.language.iso.none.fl_str_mv |
eng |
| language |
eng |
| dc.relation.references.none.fl_str_mv |
[1] A. Beilinson and V. Drinfeld. “Opers”. In: (2005). arXiv: math/0501398 [math.AG]. [2] L. Bers. “Simultaneous uniformization”. In: Bulletin of the American Mathematical Society 66.2 (1960), pp. 94–97. [3] V. G. Drinfeld and V. V Sokolov. “Lie Algebras and Equations of Korteweg-de Vries Type”. In: Journal of Mathematical Sciences 30 (1985), pp. 1975–2036. doi: 10.1007/BF02105860. [4] D. Dumas. “Complex Projective Structures”. In: (2009). arXiv: 0902.1951 [math.DG]. [5] E. Frenkel. “Gaudin model and opers”. In: (2005). arXiv: math/0407524 [math.QA]. [6] E. Frenkel. “Langlands Correspondence for Loop Groups”. In: (1992). [7] E. Frenkel. “Lectures on Wakimoto modules, opers and the center at the critical level”. In: (2002). arXiv: math/0210029 [math.QA]. [8] R. C. Gunning. Lectures on Riemann Surfaces. Princeton, NJ: Princeton University Press, 1982. [9] R. C. Gunning. “Special coordinate coverings of Riemann surfaces”. In: Math. Ann. 170 (1967), pp. 67–86. doi: 10.1007/BF01362287. [10] James E. Humphreys. Introduction to Lie Algebras and Representation Theory. Gradu-ate Texts in Mathematics, Vol. 9. New York: Springer, 1978. [11] P. Michor I. Kolar and J. lovak. Natural Operations in Differential Geometry. 1993. [12] A. W. Knapp. Lie Groups Beyond an Introduction. 2nd. Progress in Mathematics No140. Birkhäuser, 1996. [13] Anthony W. Knapp. Lie Groups Beyond an Introduction. Progress in Mathematics. Birkhäuser Boston, MA, 1996. isbn: 978-1-4757-2453-0. doi: 10.1007/978-1-4757-2453-0. [14] F. Loray and D. Marín. “Projective structures and projective bundles over compact Riemann surfaces”. In: (2007). arXiv: 0706.3608 [math.CA]. [15] W. P. Thurston. “Zippers and univalent functions”. In: 1986. doi: 10.1090/SURV/021/ |
| dc.rights.en.fl_str_mv |
Attribution-NonCommercial-NoDerivatives 4.0 International |
| dc.rights.uri.none.fl_str_mv |
http://creativecommons.org/licenses/by-nc-nd/4.0/ |
| dc.rights.accessrights.none.fl_str_mv |
info:eu-repo/semantics/openAccess |
| dc.rights.coar.none.fl_str_mv |
http://purl.org/coar/access_right/c_abf2 |
| rights_invalid_str_mv |
Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/ http://purl.org/coar/access_right/c_abf2 |
| eu_rights_str_mv |
openAccess |
| dc.format.extent.none.fl_str_mv |
47 páginas |
| dc.format.mimetype.none.fl_str_mv |
application/pdf |
| dc.publisher.none.fl_str_mv |
Universidad de los Andes |
| dc.publisher.program.none.fl_str_mv |
Matemáticas |
| dc.publisher.faculty.none.fl_str_mv |
Facultad de Ciencias |
| dc.publisher.department.none.fl_str_mv |
Departamento de Matemáticas |
| publisher.none.fl_str_mv |
Universidad de los Andes |
| institution |
Universidad de los Andes |
| bitstream.url.fl_str_mv |
https://repositorio.uniandes.edu.co/bitstreams/ea5c9d36-ef76-44a4-aecd-6470dc709509/download https://repositorio.uniandes.edu.co/bitstreams/3df8f07b-742b-4d1e-8f35-951b966d9408/download https://repositorio.uniandes.edu.co/bitstreams/95767e2b-2f6f-4de9-b547-b01de665c521/download https://repositorio.uniandes.edu.co/bitstreams/7244b742-f9a9-4da7-8ad4-409495b7648d/download https://repositorio.uniandes.edu.co/bitstreams/02945dfd-2e6d-448e-9c36-f866823f89b4/download https://repositorio.uniandes.edu.co/bitstreams/6ac08b92-0811-4d3f-b4de-52931f251181/download https://repositorio.uniandes.edu.co/bitstreams/2b9fce93-bc13-4447-acae-2ee45d6d6674/download https://repositorio.uniandes.edu.co/bitstreams/00d4185f-d96f-4258-a3aa-3c490c00f1b2/download |
| bitstream.checksum.fl_str_mv |
ae9e573a68e7f92501b6913cc846c39f 6a03f8023167d2770063cc0c2044881d e93bc5e83cf48312e8c24fc822feb42c 4460e5956bc1d1639be9ae6146a50347 70b9d0cd8b9fea878a08f58c147c9eab d65cfc8195993c532b7ca092bfd5a6c0 60104648b1506c7bb3b7faa2cbcf1713 66c3c5d0cfa9b24bd9aacc29db84e6b7 |
| bitstream.checksumAlgorithm.fl_str_mv |
MD5 MD5 MD5 MD5 MD5 MD5 MD5 MD5 |
| repository.name.fl_str_mv |
Repositorio institucional Séneca |
| repository.mail.fl_str_mv |
adminrepositorio@uniandes.edu.co |
| _version_ |
1812134020128964608 |
| spelling |
Schaffhauser, Florent Marie Rolandvirtual::321-1Cardona Guio, Alexandervirtual::322-1Aragón Rodríguez, Manuel AlejandroMalakhaltsev, Mikhailvirtual::323-12024-02-02T18:20:51Z2024-02-02T18:20:51Z2024-01-17https://hdl.handle.net/1992/73825instname:Universidad de los Andesreponame:Repositorio Institucional Sénecarepourl:https://repositorio.uniandes.edu.co/Essentially, complex projective structures arise as geometries modeled in the projective line. Meanwhile, opers appear as local systems of representation-theoretic nature and are linked to principal bundles with connections. These objects where first introduced in the punctured disk by Drinfeld and Sokolov in their study of KdV-type hierarchies and were later given a coordinate-free description for a general reductive group by Beilinson and Drinfeld. Recently, they have been studied in their connection to the geometrical Langlands correspondence and their role in the theory of vertex algebras, as developed by Frenkel. In this thesis, we study the connection between complex projective structures and opers in a geometric and algebraic way and explain how opers generalise complex projective structures when a general complex Lie group is studied.MatemáticoPregrado47 páginasapplication/pdfengUniversidad de los AndesMatemáticasFacultad de CienciasDepartamento de MatemáticasAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Opers as a generalisation of complex projective structuresTrabajo de grado - Pregradoinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_7a1fTexthttp://purl.org/redcol/resource_type/TPOpersPrincipal bundlesComplex geometryLie groupsRiemann surfacesMatemáticas[1] A. Beilinson and V. Drinfeld. “Opers”. In: (2005). arXiv: math/0501398 [math.AG].[2] L. Bers. “Simultaneous uniformization”. In: Bulletin of the American Mathematical Society 66.2 (1960), pp. 94–97.[3] V. G. Drinfeld and V. V Sokolov. “Lie Algebras and Equations of Korteweg-de Vries Type”. In: Journal of Mathematical Sciences 30 (1985), pp. 1975–2036. doi: 10.1007/BF02105860.[4] D. Dumas. “Complex Projective Structures”. In: (2009). arXiv: 0902.1951 [math.DG].[5] E. Frenkel. “Gaudin model and opers”. In: (2005). arXiv: math/0407524 [math.QA].[6] E. Frenkel. “Langlands Correspondence for Loop Groups”. In: (1992).[7] E. Frenkel. “Lectures on Wakimoto modules, opers and the center at the critical level”. In: (2002). arXiv: math/0210029 [math.QA].[8] R. C. Gunning. Lectures on Riemann Surfaces. Princeton, NJ: Princeton University Press, 1982.[9] R. C. Gunning. “Special coordinate coverings of Riemann surfaces”. In: Math. Ann. 170 (1967), pp. 67–86. doi: 10.1007/BF01362287.[10] James E. Humphreys. Introduction to Lie Algebras and Representation Theory. Gradu-ate Texts in Mathematics, Vol. 9. New York: Springer, 1978.[11] P. Michor I. Kolar and J. lovak. Natural Operations in Differential Geometry. 1993.[12] A. W. Knapp. Lie Groups Beyond an Introduction. 2nd. Progress in Mathematics No140. Birkhäuser, 1996.[13] Anthony W. Knapp. Lie Groups Beyond an Introduction. Progress in Mathematics. Birkhäuser Boston, MA, 1996. isbn: 978-1-4757-2453-0. doi: 10.1007/978-1-4757-2453-0.[14] F. Loray and D. Marín. “Projective structures and projective bundles over compact Riemann surfaces”. In: (2007). arXiv: 0706.3608 [math.CA].[15] W. P. Thurston. “Zippers and univalent functions”. In: 1986. doi: 10.1090/SURV/021/201923451Publicationhttps://scholar.google.es/citations?user=Xu_2niIAAAAJvirtual::321-10000-0002-9511-1760virtual::321-1https://scienti.minciencias.gov.co/cvlac/visualizador/generarCurriculoCv.do?cod_rh=0001495730virtual::321-1https://scienti.minciencias.gov.co/cvlac/visualizador/generarCurriculoCv.do?cod_rh=0000555789virtual::323-1https://scienti.minciencias.gov.co/cvlac/visualizador/generarCurriculoCv.do?cod_rh=0000055190virtual::322-1ce3016b4-e0fe-47c4-8953-f7fbc22fa01avirtual::321-1b65b9b87-c23b-4157-ac5a-55f34b071dc7virtual::322-1ce3016b4-e0fe-47c4-8953-f7fbc22fa01avirtual::321-1b65b9b87-c23b-4157-ac5a-55f34b071dc7virtual::322-1680ba16d-3876-4cbe-ab2b-9b0a0964eadbvirtual::323-1680ba16d-3876-4cbe-ab2b-9b0a0964eadbvirtual::323-1LICENSElicense.txtlicense.txttext/plain; charset=utf-82535https://repositorio.uniandes.edu.co/bitstreams/ea5c9d36-ef76-44a4-aecd-6470dc709509/downloadae9e573a68e7f92501b6913cc846c39fMD51ORIGINALFormato_Repositorio.pdfFormato_Repositorio.pdfHIDEapplication/pdf452810https://repositorio.uniandes.edu.co/bitstreams/3df8f07b-742b-4d1e-8f35-951b966d9408/download6a03f8023167d2770063cc0c2044881dMD53Opers as a generalisation of complex projective structures.pdfOpers as a generalisation of complex projective structures.pdfapplication/pdf366547https://repositorio.uniandes.edu.co/bitstreams/95767e2b-2f6f-4de9-b547-b01de665c521/downloade93bc5e83cf48312e8c24fc822feb42cMD54CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8805https://repositorio.uniandes.edu.co/bitstreams/7244b742-f9a9-4da7-8ad4-409495b7648d/download4460e5956bc1d1639be9ae6146a50347MD55TEXTFormato_Repositorio.pdf.txtFormato_Repositorio.pdf.txtExtracted texttext/plain1999https://repositorio.uniandes.edu.co/bitstreams/02945dfd-2e6d-448e-9c36-f866823f89b4/download70b9d0cd8b9fea878a08f58c147c9eabMD56Opers as a generalisation of complex projective structures.pdf.txtOpers as a generalisation of complex projective structures.pdf.txtExtracted texttext/plain103208https://repositorio.uniandes.edu.co/bitstreams/6ac08b92-0811-4d3f-b4de-52931f251181/downloadd65cfc8195993c532b7ca092bfd5a6c0MD58THUMBNAILFormato_Repositorio.pdf.jpgFormato_Repositorio.pdf.jpgGenerated Thumbnailimage/jpeg11287https://repositorio.uniandes.edu.co/bitstreams/2b9fce93-bc13-4447-acae-2ee45d6d6674/download60104648b1506c7bb3b7faa2cbcf1713MD57Opers as a generalisation of complex projective structures.pdf.jpgOpers as a generalisation of complex projective structures.pdf.jpgGenerated Thumbnailimage/jpeg8087https://repositorio.uniandes.edu.co/bitstreams/00d4185f-d96f-4258-a3aa-3c490c00f1b2/download66c3c5d0cfa9b24bd9aacc29db84e6b7MD591992/73825oai:repositorio.uniandes.edu.co:1992/738252024-02-16 15:23:13.343http://creativecommons.org/licenses/by-nc-nd/4.0/Attribution-NonCommercial-NoDerivatives 4.0 Internationalopen.accesshttps://repositorio.uniandes.edu.coRepositorio institucional Sénecaadminrepositorio@uniandes.edu.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 |