Appears in collection : Algèbre, Géométrie et Physique : une conférence en l'honneur

We study some mathematical models of Penrose's idea to interpret 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». We suggest to model the kinematics of Big Bang as an algebraic geometric (or analytic) blow up of a point at the boundary of the next aeon space--time. Moreover, we argue that at the moment of crossover (infinite future of the previous aeon) time undergoes the Wick rotation and becomes purely imaginary on the crossover boundary. The Mixmaster model of the early history of the next aeon is neatly includedinto this picture by postulatingthat the reverse Wick rotation follows a hyperbolic geodesic connecting imaginary time axis to the real one.