Evolution of geometric ideas, and the role of Relativity

The idea of space for a working mathematician in the 21st century is very fluid. So much so that it has become difficult to imagine the magnitude of the psychological resistance people had to overcome over the last two centuries. In this talk, I would like to summarize the evolution of the geometric ideas that leads to our current understanding of the universe, tracing the trajectory of ideas provided by Euclid, Newton, Einstein, Minkowski, Weyl, and more recently Hawking and Penrose. In doing so, it is inevitable to access the meaning of the Theory of Relativity. Despite being a physical theory, the geometry implied by the physical structure of the relativistic universe is so simple yet so rich mathematically, independent of the physical contexts. At this point of our scientific endeavor, where the geometry of the spacetime advocated by Einstein has been for the first time objectively verified via gravitational wave detection, it is only natural to ponder on the long and winding road of geometric ideas.