String Theory and the Geometry of the Universe’s Hidden Dimensions Shing-Tung Yau and Steve Nadis gone through a radical This article is adapted from a talk that Shing-Tung change in the nineteenth Yau gave at the University of California, Berkeley, on February 10, 2011. The talk was written by Yau century, thanks to the and Nadis, based on their book, The Shape of Inner contributions of Gauss Space (Basic Books, 2010). Both the talk and the book and Riemann. Riemann are presented from Yau’s perspective, with the story revolutionized our no- told in Yau’s voice. tions of space, freeing up mathematics in the ’d like to talk about how mathematics and process. physics can come together to the benefit Objects no longer of both fields, particularly in the case of had to be confined to Calabi-Yau spaces and string theory. This, not the flat, linear space of coincidentally, is the subject of the new book Euclidean geometry. Rie- I mann instead proposed I have coauthored, The Shape of Inner Space. This book tells the story of those spaces. It also tells a much more abstract some of my own story and a bit of the history of conception of space—of any possible dimension— geometry as well. In that spirit, I’m going to back in which we could describe distance and curvature. up and talk about my personal introduction to In fact, one can develop a form of calculus that is geometry and the evolution of the ideas that are especially suited to such an abstract space. discussed in this book. About fifty years later, Einstein realized that this kind of geometry, which involved curved I wanted to write this book to give people a spaces, was exactly what he needed to unify Newto- sense of how mathematicians think and approach nian gravity with special relativity. This insight cul- the world. I also want people to realize that math- minated in his famous theory of general relativity. ematics does not have to be a wholly abstract dis- I learned about Riemannian geometry during my cipline, disconnected from everyday phenomena, first year at Berkeley in 1969. It was different from but is instead crucial to our understanding of the the classical geometry that I studied in college in physical world. Hong Kong, where we focused on curves and sur- So we’re now going to step back a bit in time. faces in linear space. At Berkeley, I took courses Or perhaps I should say step back in spacetime… from Spanier on algebraic topology, Lawson on Riemannian geometry, and Morrey on partial Riemannian Geometry differential equations. I also audited courses on When I arrived in Berkeley in 1969 for graduate many other subjects, including general relativity, study, I learned that the concept of geometry had taking in as much information as I could possibly Shing-Tung Yau is the William Casper Graustein Professor assimilate. of Mathematics at Harvard University. His email address Algebraic topology was rather new to me. But, is [email protected]. after a couple of months, I was able to understand Steve Nadis, a writer based in Cambridge, Massachusetts, what a fundamental group is, while also picking is the coauthor (with Yau) of The Shape of Inner Space up some elementary facts about homotopy and (Basic Books, 2010). His email address is stevenadis@ homology theory. comcast.net. SEPTEMBER 2011 NOTICES OF THE AMS 1067 There were about five hundred graduate stu- dents in the mathematics department at the time, and nobody had an office. We were in Campbell Hall, and instructors used the T building—the wood building in front of Evans Hall. I spent all my free time in the mathematics li- Figure 1. In topology, a sphere, cube, square brary, which served as my unofficial office, where pyramid, and tetrahedron—among other I constantly searched for interesting articles to shapes—are all considered equivalent. [Xianfeng pass the time. During the Christmas holiday of (David) Gu and Xiaotian (Tim) Yin in The Shape of that year, when everybody else went home, I read Inner Space.] John Milnor’s paper in the Journal of Differential Geometry1 on the relation of the fundamental group to the curvature of a manifold. I found that exciting because those were exactly the concepts that I had just studied. And Milnor was such a good writer that I understood everything in his paper. He referred to another paper by Preissman2 that sounded interesting to me. From these papers, I learned that if the space has negative curvature, there is a strong constraint on the “fundamental group”—a concept from topology. Such a group consists of closed loops in that space, each of which has an initial point fixed. The elements of this group, which can be deformed to each other, are considered equivalent. Preissman’s theorem says that, in the fundamen- tal group of manifolds with negative curvature, every two commuting elements can be written as a multiple of some other element in the group. This was intriguing, and I started to toy around Figure 2. In topology, there are just two kinds of with Preissman’s paper, trying to see what would one-dimensional spaces that are fundamentally happen if the space is allowed to have nonpositive different from one another: a line and a circle. curvature. This was the first time I got into state- Two-dimensional (orientable) surfaces can ments linking the curvature of a space—a precise be classified by their genus or number of description of the geometry—to a much cruder, holes. A sphere of genus 0, with no holes, is more general way of characterizing shape, which fundamentally distinct from a donut of genus we call topology. 1, which has one hole. [Xianfeng (David) Gu and Topology is a concept of a space that is unre- Xiaotian (Tim) Yin in The Shape of Inner Space.] lated to the way that we measure distance in that space. In that sense, topology describes a space much less precisely than geometry does. We need to know all the details of a space to measure the distance between any two points. The sum of all those details, which spell out the curvature at every point, is what we mean by geometry (Figures 1, 2, 3). A donut and a coffee mug, for example, have the same topology, but they have a different shape or geometry. A sphere and an ellipsoid, similarly, have the same topology, but they generally have a different shape. The sphere is a topological space with no fundamental group, as every closed loop can be shrunk to a point continuously. But there 1J. Milnor, A note on curvature and fundamental group, Figure 3. The donut has the same topology J. Differential. Geometry 2 (1968), 1–7. as this less familiar (and perhaps less tasty) 2A. Preissman, Quelques propriétés globales des espaces object, but they have a different shape (or de Riemann, Comment. Math. Helv. 15 (1942–1943), geometry). [Courtesy of Xiaotian (Tim) Yin.] 175–216. 1068 NOTICES OF THE AMS VOLUME 58, NUMBER 8 are closed curves on the surface of a torus that special theory of relativity, establishing the equiva- cannot be shrunk to a point continuously. lence of so-called inertial frames of reference, I wrote down my generalization of Preissman’s which he presented in 1905. A couple of years theorem, which links topology to geometry.3 later, he realized that gravity could not be treated While I was photocopying those notes in the within special relativity, which was a linear theory, Xerox room, I ran into Arthur Fisher, a mathemati- but instead needed to be treated in a separate, cal physicist. He insisted on knowing what I had nonlinear theory. He then began to work on the written. After reading through my notes, he told latter theory, which came to be known as general me that any principle that related curvature with relativity, admitting that “it took me a long time to topology would be useful in physics. His comments see what coordinates at all meant in physics.” The have stayed with me ever since. notion of equivalence, which held that the laws of gravity should be true in any coordinate system, General Relativity had been his guiding principle. By 1912, he started We learned through special relativity that space to realize that the gravitational potential should be and time should not be treated separately but described by a second-order symmetric tensor—a should instead be merged together to form Riemannian metric with a Lorentzian signature.5 spacetime. Einstein struggled in his attempt to Two additional problems had to be solved as obtain a fundamental description of gravity. But he well, Einstein noted: 1. How can a field law, ex- got some help from his friend Marcel Grossman, a pressed in terms of the special theory of relativity, mathematician, who told him of the work of other be transferred to the case of a Riemannian metric? mathematicians, Riemann and Ricci. 2. What are the laws that determine the Rieman- Riemann provided the framework of abstract nian metric itself?6 space, as well as the means for defining distance He worked on these problems from 1912 to and curvature in such a space. Riemann thus sup- 1914 with Grossman. Together they determined plied the background space or setting in which that the mathematical methods for solving the first gravity, as Einstein formulated it, plays out. problem could be found in the differential calculus But Einstein also drew on the work of Ricci, of Ricci and Levi-Civita.