The Shape of Space: Eugenio Calabi (1923-2023)

Eugenio Calabi (May 1923 – Sep 2023). Source: Quanta Magazine.

Eugenio Calabi (1923-2003), who died last month at the age of 100, was regarded as one of the greatest geometers of the 20th century. Among his many inventive accomplishments is the Calabi-Yau manifold, a mathematical object of exquisite beauty which has found applications in an esoteric branch of physics known as string theory (more of that later).

Calabi was born in Milan, Italy in 1923. He did not have set out to become a mathematician though his talent for numbers showed up early. In 1939, he decided to major in chemical engineering when he arrived at the Massachusetts Institute of Technology after his family fled Italy at the outset of World War II. He was only as 16 years old at the time. During the war, he served as a U.S. Army translator in France and Germany. After he returned home, he worked briefly as a chemical engineer before deciding to switch to math, graduating with a doctorate at Princeton. He then held a series of professorships before landing at University of Pennsylvania in 1964, where he would remain until his death in September 2023.

In the 1970s, physicists were trying to devise theories to unify the four fundamental forces of nature, which led to to the idea that all fundamental particles such as electrons are composed of exceedingly tiny vibrating strings, with different patterns of vibration manifesting as different particles. This string theory of vibrations only work out correctly in 10 dimensions, six more than the three spatial dimensions we are accustomed to, plus a fourth dimension, time.

For a while, Calabi had been studying something called Kähler manifolds, named after the 20th-century German geometer Erich Kähler. A manifold is a surface with many dimensions that looks “locally flat” at every point, like the way the curved surface of the Earth appears as flat in most of our everyday experience. Kähler manifolds are smooth, meaning that they have no sharp or jagged features, and they only come in even dimensions — 2, 4, 6 and up.

Calabi wasn’t studying manifolds because they were useful; he just thought that doing math was fun. Nevertheless, his explorations would soon find its way into one of the deepest areas in theoretical physics – string theory. The landmark year was 1977 when a 28-year old Chinese mathematician named Shing-tung Yau (b. 1949) burst into the scene and proved a conjecture of Calabi that Kahler manifolds should have “simple curvatures” that respect a certain kind of smoothness. Yau’s proof confirmed Calabi’s intuition, and he was awarded the Fields Medal, the mathematical equivalent of a Nobel Prize for this work. The manifold that Calabi conjectured and Yau proved has since been known as Calabi-Yau manifold. I mentioned earlier that the Calabi-Yau manifold is an object of exquisite beauty. This is how it looks like in cross-section:

The cross-section of a Calabi-Yau manifold gives a glimpse into its complexity. Photo: vchal/Shutterstock.

By the mid-1980s, a group of physicists had realized that the six “extra” dimensions of the universe might be “folded up” in a minute Calabi manifold and kept out of sight from our everyday experience, so that the world continues to appear to us as having only three spatial dimensions plus time.

It isn’t clear if our world is described by string theory, or that the extra dimensions posited by string theory are secretly filled with Calabi-Yau manifold, but it is the best shot at a “Theory of Everything” that physicists and mathematicians have been able to cobble up so far.

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