Time Between Real and Imaginary: what Geometries Describe Universe near Big Bang?

Abstract

For about a century, a great challenge for theoretical physics consisted in understanding the role of quantum mode of description of our Universe (\quantum gravity"). Einstein space{times on the scale of observable Universe do not easily submit to any naive quantization scheme. There are better chances to concoct a satisfying quantum picture of the very early space{time, near the Big Bang, where natural scales of events like in ation extrapolated from current observations resist any purely classical description and rather require quantum input. Many physicists and mathematicians tried to understand the quantum early Universe, sometimes unaware of input of the other community. One of the goals of this article is to contribute to the communication of the two communities. In the main text, I present some ideas and results contained in the recent survey/research papers [Le13] (physicists) and [MaMar14], [MaMar15] (mathematicians).