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 SIGMA, 2014, Volume 10, 073, 20 pages (Mi sigma938)

Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology

Yuri I. Manina, Matilde Marcollib

a Max-Planck-Institut für Mathematik, Bonn, Germany
b California Institute of Technology, Pasadena, USA

Abstract: 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.

Keywords: Big Bang cosmology; algebro-geometric blow-ups; cyclic cosmology; Mixmaster cosmologies; modular curves.

DOI: https://doi.org/10.3842/SIGMA.2014.073

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ArXiv: 1402.2158
MSC: 85A40; 14N05; 14G35
Received: March 1, 2014; in final form June 27, 2014; Published online July 9, 2014
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Citation: Yuri I. Manin, Matilde Marcolli, “Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology”, SIGMA, 10 (2014), 073, 20 pp.

Citation in format AMSBIB
\Bibitem{ManMar14} \by Yuri~I.~Manin, Matilde~Marcolli \paper Big Bang, Blowup, and Modular Curves: Algebraic Geometry in~Cosmology \jour SIGMA \yr 2014 \vol 10 \papernumber 073 \totalpages 20 \mathnet{http://mi.mathnet.ru/sigma938} \crossref{https://doi.org/10.3842/SIGMA.2014.073} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000339447300001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84904180237} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Chanda S., Guha P., Roychowdhury R., “Bianchi-IX, Darboux–Halphen and Chazy–Ramanujan”, Int. J. Geom. Methods Mod. Phys., 13:4 (2016), 1650042
2. Manin, Y.; Marcolli, M., “Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies”, Annales de la faculte des sciences de Toulouse Ser. 6, 25:2-3 (2016), 517-542
3. Manin, Y. I., “Painlevé VI equations in p-adic time”, P-Adic Numbers, Ultrametric Analysis, and Applications, 8:3 (2016), 217-224
4. B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, E. I. Zelenov, “P-adic mathematical physics: the first 30 years”, P-Adic Numbers Ultrametric Anal. Appl., 9:2 (2017), 87–121
5. M. Marcolli, “Spectral action gravity and cosmological models”, C. R. Phys., 18:3-4 (2017), 226–234
6. P. Gallardo, N. Giansiracusa, “Modular interpretation of a non-reductive Chow quotient”, Proc. Edinb. Math. Soc., 61:2 (2018), 457–477
7. W. Fan, F. Fathizadeh, M. Marcolli, “Motives and periods in Bianchi IX gravity models”, Lett. Math. Phys., 108:12 (2018), 2729–2747
8. Fan W., Fathizadeh F., Marcolli M., “Modular Forms in the Spectral Action of Bianchi Ix Gravitational Instantons”, J. High Energy Phys., 2019, no. 1, 234
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