22 / Two-dimensional manifolds are also known as surfaces. A sphere (left) is positively curved, while the surface on the right, which resembles a six-connection pipe fitting, is negatively curved. Reprinted with permission from Gerard Westendorp. /

dimensional space-time. A long list of profound discoveries followed from the equations that Einstein wrote down in 1915: black holes, the expanding universe, the big bang, and dark energy. To understand any of these ideas fully, you have to learn Riemannian geometry. Somewhere, Galileo must be smiling.

But Einstein’s general relativity was only the beginning. Similar geometric constructions underlie the field theories that describe particle physics. The discovery of antimatter, in 1932, grew directly out of an attempt to reconcile relativity with the quantum-mechanical description of the electron. The equations predicted extra solutions that seemed like positively charged electrons. We now call them positrons. They are the key ingredients in positron emission tomography, or PET scans, which are used to study the workings of the human brain.

In the later 1930s and 1940s, physicists and mathematicians started losing touch with one another. Physicists started thinking about fields that permeate all of space, which they called “gauge fields.” (Examples include the electromagnetic field and the weak and strong nuclear forces.) Meanwhile mathematicians, for different reasons, became very interested in a new kind of geometric space, called a fiber bundle, which is roughly like a curved space with a quiver of arrows attached at every point (see Figure 23 for an example). It wasn’t until the 1970s that mathematicians and physicists realized that they were doing the same thing. The physicists’ gauge fields were like individual arrows in the mathematicians’ quiver of arrows.

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