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

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).