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

This article is cited in 8 scientific papers (total in 8 papers)

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.


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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

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
\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

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    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath
    3. Manin, Y. I., “Painlevé VI equations in p-adic time”, P-Adic Numbers, Ultrametric Analysis, and Applications, 8:3 (2016), 217-224  crossref  mathscinet  zmath
    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  crossref  mathscinet  zmath  isi  scopus
    5. M. Marcolli, “Spectral action gravity and cosmological models”, C. R. Phys., 18:3-4 (2017), 226–234  crossref  isi  scopus
    6. P. Gallardo, N. Giansiracusa, “Modular interpretation of a non-reductive Chow quotient”, Proc. Edinb. Math. Soc., 61:2 (2018), 457–477  crossref  mathscinet  isi
    7. W. Fan, F. Fathizadeh, M. Marcolli, “Motives and periods in Bianchi IX gravity models”, Lett. Math. Phys., 108:12 (2018), 2729–2747  crossref  mathscinet  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
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