Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology
We introduce some algebraic geometric models in cosmology related to the ''boundaries'' of space-time: Big Bang, Mixmaster Universe, Penrose's crossovers between aeons. We suggest to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a po...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2014 |
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Інститут математики НАН України
2014
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/146619 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology / Y.I. Manin, M. Marcolli // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 50 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862546436333240320 |
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| author | Manin, Y.I. Marcolli, M. |
| author_facet | Manin, Y.I. Marcolli, M. |
| citation_txt | Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology / Y.I. Manin, M. Marcolli // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 50 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We introduce some algebraic geometric models in cosmology related to the ''boundaries'' of space-time: Big Bang, Mixmaster Universe, Penrose's crossovers between aeons. We suggest to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a point x. This creates a boundary which consists of the projective space of tangent directions to x and possibly of the light cone of x. We argue that time on the boundary undergoes the Wick rotation and becomes purely imaginary. The Mixmaster (Bianchi IX) model of the early history of the universe is neatly explained in this picture by postulating that the reverse Wick rotation follows a hyperbolic geodesic connecting imaginary time axis to the real one. Penrose's idea to see the Big Bang as a sign of crossover from ''the end of previous aeon'' of the expanding and cooling Universe to the ''beginning of the next aeon'' is interpreted as an identification of a natural boundary of Minkowski space at infinity with the Big Bang boundary.
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| first_indexed | 2025-11-25T12:50:12Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-146619 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-25T12:50:12Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
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| spelling | Manin, Y.I. Marcolli, M. 2019-02-10T10:10:16Z 2019-02-10T10:10:16Z 2014 Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology / Y.I. Manin, M. Marcolli // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 50 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 85A40; 14N05; 14G35 DOI:10.3842/SIGMA.2014.073 https://nasplib.isofts.kiev.ua/handle/123456789/146619 We introduce some algebraic geometric models in cosmology related to the ''boundaries'' of space-time: Big Bang, Mixmaster Universe, Penrose's crossovers between aeons. We suggest to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a point x. This creates a boundary which consists of the projective space of tangent directions to x and possibly of the light cone of x. We argue that time on the boundary undergoes the Wick rotation and becomes purely imaginary. The Mixmaster (Bianchi IX) model of the early history of the universe is neatly explained in this picture by postulating that the reverse Wick rotation follows a hyperbolic geodesic connecting imaginary time axis to the real one. Penrose's idea to see the Big Bang as a sign of crossover from ''the end of previous aeon'' of the expanding and cooling Universe to the ''beginning of the next aeon'' is interpreted as an identification of a natural boundary of Minkowski space at infinity with the Big Bang boundary. This paper is a contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in
 honor of Marc A. Rief fel. The full collection is available at http://www.emis.de/journals/SIGMA/Rieffel.html. 
 This paper was conceived after the lecture in Bonn (November 2013), in which Sir Roger Penrose
 explained his fascinating ideas about cyclic cosmology. Ya. Sinai and O. Bogoyavlenskii made
 helpful remarks about BKLL treatment of the Bianchi IX model. We are grateful to them. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology Article published earlier |
| spellingShingle | Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology Manin, Y.I. Marcolli, M. |
| title | Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology |
| title_full | Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology |
| title_fullStr | Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology |
| title_full_unstemmed | Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology |
| title_short | Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology |
| title_sort | big bang, blowup, and modular curves: algebraic geometry in cosmology |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146619 |
| work_keys_str_mv | AT maninyi bigbangblowupandmodularcurvesalgebraicgeometryincosmology AT marcollim bigbangblowupandmodularcurvesalgebraicgeometryincosmology |