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Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology
Репозиторій DSpace/Manakin
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2014, том 10
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Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology
Інший ID:
2010 Mathematics Subject Classification: 85A40; 14N05; 14G35
DOI:10.3842/SIGMA.2014.073
DOI:10.3842/SIGMA.2014.073
Посилання:
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.
Підтримка:
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.
Дата:
2014
Переглядів:
548
Завантажень:
273
Анотація:
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.
