String theory has its scientific origins in the late 1970s and early 1980s, but it was propelled into the full view of the public in 2000 thanks to Brian Greene's readily accessible and scientifically accurate (if mathematically devoid) prose in The Elegant Universe. In the intervening decade, the basic ideas of string theory—that the Universe is potentially made up of little strings—has become fairly well known by members of the public. However, with no real possible experimental test to directly probe for the existence of these tiny strings, many within and outside the scientific community question the validity of a "theory of everything" that has no readily testable predictions.

One thing that, in my experience, constantly gets overlooked is that, in order to pursue the study of string theory, entire fields of mathematics have been revolutionized and brought back from what was the dust bin of mathematical curiosity. Their resurgence and renewed interest exist completely independently of whether string theory is right or if it gets left alongside Ptolemaic epicycles in the annals of scientific ideas.

This is a point that is driven at—although concretely mentioned too late, in my opinion—in The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions by Shing-Tung Yau and Steve Nadis. Anyone with even a passing familiarity with string theory should recognize the name Yau as the latter half of the ubiquitous Calabi-Yau manifold that appears throughout string theory. The book recounts Yau's experience with how he came to solve the Calabi conjecture, and how his proof was later rediscovered by physicists to become the cornerstone in a new theory researchers were playing with.

The book starts with the beginnings of geometry and its influence on physics by first discussing the Platonic solids and how the ancient Greeks viewed them in relation to the physical properties of the real world. It then jumps forward a few thousand years, to where Yau briefly mentions his childhood in China and his budding interest in geometry. Since the book is not an autobiography, it quickly shifts back to the mathematics that Yau has spent his career studying, including topology, algebraic geometry, and geometric analysis—the tools that laid the foundations for his later work.

Yau doesn't hold the reader's hand in the book, and he expects the reader to have an intuition capable of understanding some non-obvious mathematical and geometric ideas. For instance, the product of two S1 manifolds (circles in a plane) is a rectangle rolled up on itself twice (think about it for a minute). The book gets down to the core of its focus when Yau and Nadis introduce the Calabi conjecture, the geometric problem that gave rise to Calabi-Yau manifolds. Calabi's question boils down to this:

Can a compact Kähler manifold with a vanishing first Chern class also have a Ricci-flat metric?

How this turned the world of physics on its head is hard to see at first, but Yau explains this conjecture in more physical terms. The question, while purely mathematical in nature, can be intimately tied to Einstein's field equations for general relativity. Viewed through the lens of general relativity, this question could be rephrased as, "Could there be gravity in our Universe even if space was totally devoid of matter?" As Yau puts it, if Calabi was correct, then curvature alone would make gravity possible, even in the absence of matter.

The next section of the book focuses on Yau's solution to the Calabi conjecture, and it raises an interesting point that highlights the difference between mathematicians and scientists/engineers. Yau's proof to the Calabi conjecture in 1976 (published in two papers in 1977 and 1978) showed that a manifold that satisfied Calabi's requirements (compact, Kähler, Ricci-flat) exists... nothing more. Nothing about the nature of it, what it looks like, or if a solution in the form of a realization can actually be computed. As an engineer this doesn't help me one bit, but to a mathematician, as Yau puts it, the fact that an answer exists is the answer. There is no need to go further. It reminds me of an old joke:

In three separate locked, airtight, rooms sit a physicist, an engineer, and a mathematician. In each room is a sealed box with a key that will allow escape. Upon learning of their predicament, all get to working on the problem at hand. The physicist works out the material properties and forces needed to exactly pry the box open with a small screwdriver, applies the minimal amount of force needed, gets the key, and leaves. The engineer takes the brute force approach: he smashes the box to pieces, picks up the key, opens the door, and leaves. The mathematician is not heard from for weeks, and upon going back to look for him, the engineer and physicist find him dead in the room. In front of him are pages upon pages of a mathematical proof. At the end it is written, "a solution exists."

It wasn't until eight years later, in 1984, that physicists started realizing the importance of manifolds such as those described by Calabi-Yau in the study of string theory. As Yau explains it, it was the need for supersymmetry that lead to the mathematical concept of holonomy, which in turn lead to Calabi-Yau manifolds. Once the connection was made, physicists started creating actual realizations of Calabi-Yau manifolds and studying the properties of these spaces and whether or not they could predict the properties of the universe we see around us.

The remainder of the book discusses how the mathematics has advanced thanks to work of physicists and how physics has advanced thanks to the work of mathematicians. While the book focuses heavily on the mathematics, it is only in one of the last chapters where Yau really hammers his key point home: no matter what happens with string theory, the mathematics developed to describe it will still be correct and will form the foundation for future generations of mathematicians and geometers. The Shape of Inner Space proves to be an interesting read to anyone with an interest not only in string theory, but in the mathematics that underlie this potential theory of everything.

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