On space-time structure and the Universe: some issues of concept and principle

In this discourse, we would like to discuss some issues of concept and principle in the context of the following three aspects. One, how [Formula: see text] arises as a constant of space-time structure on the same footing as the velocity of light. These are the two constants innate to space-time without reference to any force or dynamics whatsoever, and are interwoven in the geometry of 'free' homogeneous space-time. Two, how does the vacuum energy gravitate? Could its gravitational interaction in principle be included in general relativity or a new theory of quantum space-time/gravity would be required? Finally, we would like to raise the fundamental question: How does the Universe physically expand? Since there does not lie anything outside into which it can expand, instead it has to expand on its own-maybe by creating new space-time out of nothing at each instant and at every location! Thus not only was the Universe created at some instant in the past marking the beginning in the Big Bang, it is in fact being created continuously at each epoch as it expands. We thus need quantum theory of space-time/gravity for fully understanding the working of the Universe. This article is part of the theme issue 'The future of mathematical cosmology, Volume 2'.