I've been reading this novel lately written by the Chair of the Harvard Mathematics department, Shing-Tung Yau, titled The Shape of Inner Space. So far, it is primarily on Geometry and how it relates to a particular area of physics called String Theory. String Theory says that space is governed by some sort of objects called strings that wiggle and twist and determine the laws of the universe. These strings can be described using an area of Geometry called Differential Geometry. The Geometry learned in high schools is called Euclidean Geometry, which deals with straight lines and measurements on space that is flat. Differential Geometry deals with measurements on space that is curved, such as the Earth. It turns out that Differential Geometry is essential to the study of Physics, because at the beginning of the 19th century Albert Einstein showed that space, or rather spacetime, is itself curved. Differential Geometry has been applied many times to the study of Physics, and is currently being applied right now to study String Theory. One of the consequences of String Theory, if it is true, is that spacetime is actually ten dimensional, instead of the four dimensional point of view proposed by Einstein. The ten dimensions consist of the four dimensions of spacetime, plus an additional six dimensions so tiny that we will most likely never see them. These six dimensions are stored in shapes known as Calabi-Yau manifolds, named after their discoverers, one of which is the author of this book. Essentially what a manifold is a shape that in tiny little regions, looks "flat". The surface of the Earth is roughly a manifold (not considering mountains or steep valleys), because to a person standing at the surface, it simply looks like a plane. Similarly, manifolds can be extended to higher dimensions, where locally they look like "flat" space, but globally, they could look entirely different.
Sadly, since these Calabi-Yau manifolds are said to be so miniscule, we will probably never be able to perform experiments to determine if String Theory holds or not. However, because of String Theory, we have been able to study Geometry from completely different points of view, which has helped advance the current field of Geometry as we know it.
