Geometry: from Riemann to Einstein and on to String Theory

In this talk, the concept of how space has evolved in the past one hundred fifty years since the time of Riemann will be discussed. Even some old ideas, such as the Riemann mapping theorem, are now currently being used in computer graphics. Some of the applications will be demonstrated. The ideas of Riemann were used, successfully, by Einstein in formulating new ideas about gravity that com- bined Newtonian theory with special relativity. In his theory of general relativity, Einstein generalized the notion of space to a ‘spacetime’ of four dimensions. The idea of unifying all the forces in nature was pursued by Einstein and continued by string theorists, who took physics from a four-dimensional spacetime to a ten- dimensional spacetime. The six extra dimensions, which are posited by string theo- ry, can be called the ‘inner space’ of the universe, and the research that explores this hidden (and hypothetical) realm shall be reported. Much of this work is dis- cussed in Prof. Yau’s book with Steve Nadis, The Shape of Inner Space.