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intuitive reasoning here is that as a string moves it can sweep across the compressed chunk of space, like a hula hoop across a soap bubble, and thus act as an encircling protective barrier. The calculations showed that such a “string shield” nullifies any potentially disastrous consequences, ensuring that string theory’s equations suffer no ill effect—no “1 divided by 0” type errors—even though the equations of conventional general relativity would fall apart.

In the years since, researchers have shown that a variety of other more complicated singularities (with names like conifolds, orientifolds, enhancons …) are also under full control within string theory. So there’s a growing list of situations that would have left Einstein, Bohr, Heisenberg, Wheeler, and Feynman saying, “We just don’t know what’s going on,” and yet for which string theory gives a complete and consistent description.

This is great progress. But a remaining challenge for string theory is to cure the singularities of black holes and the big bang, which are more severe than those so far addressed. Theorists have expended much effort trying to reach this goal, and they’ve taken significant strides. But the executive summary is that there is still a way to go before these most puzzling and most relevant of singularities are fully understood.

Nevertheless, one major advance has illuminated a related aspect of black holes. As I will discuss in Chapter 9, the work of Jacob Bekenstein and Stephen Hawking in the 1970s established that black holes contain a very particular quantity of disorder, technically known as entropy. According to basic physics, much as the disorder within a sock drawer reflects the many possible haphazard rearrangements of its contents, the disorder of a black hole reflects the many possible haphazard rearrangements of the black hole’s innards. But try as they might, physicists were unable to understand black holes well enough to identify their innards, let alone analyze the possible ways they could be rearranged. The string theorists Andrew Strominger and Cumrun Vafa broke through the impasse. Using a mélange of string theory’s fundamental ingredients (some of which we will encounter in Chapter 5), they created a mathematical model for a black hole’s disorder, a model transparent enough to enable them to extract a numerical measure of the entropy. The result they found agreed spot-on with the Bekenstein-Hawking answer. While the work left open many deep issues (such as explicitly identifying a black hole’s microscopic constituents), it provided the first firm quantum mechanical accounting of a black hole’s disorder.16

The remarkable advances in dealing with both singularities and black hole entropy give the community of physicists well-grounded confidence that in time the remaining challenges of black holes and the big bang will be conquered.

String Theory and Mathematics

Making contact with data, experimental or observational, is the only way to determine if string theory correctly describes nature. It’s a goal that’s proved elusive. String theory, for all its advances, is still a wholly mathematical undertaking. But string theory isn’t just a consumer of math. Some of its most important contributions have been to mathematics.

When he was developing the general theory of relativity in the early twentieth century, Einstein famously mined the mathematical archives in search of rigorous language for describing curved spacetime. The earlier geometrical insights of mathematicians such as Carl Friedrich Gauss, Bernhard Riemann, and Nikolai Lobachevsky provided an important foundation for his success. In a sense, string theory is now helping to repay Einstein’s intellectual debt by driving the development of new mathematics. There are numerous examples, but let me give one that captures the flavor of string theory’s mathematical achievements.

General relativity established a tight link between the geometry of spacetime and the physics we observe. Einstein’s equations, together with the distribution of matter and energy in a region,