# The Mathematician Who Sculpted the Shape of Space

## Introduction

Eugenio Calabi was known to his colleagues as an inventive mathematician — “transformatively original,” as his former student Xiuxiong Chen put it. In 1953, Calabi began to contemplate a class of shapes that nobody had ever envisioned before. Other mathematicians thought their existence was impossible. But a couple of decades later, these same shapes became extremely important in both math and physics. The results ended up having a far broader reach than anyone, including Calabi, had anticipated.

Calabi was 100 years old when he died on September 25, mourned by his colleagues as one of the most influential geometers of the 20th century. “A lot of mathematicians like to solve problems that finish off work on a particular subject,” Chen said. “Calabi was someone who liked to start a subject.”

Jerry Kazdan, who taught with Calabi at the University of Pennsylvania for nearly 60 years, said that his colleague “had a special way of looking at things. Taking the less obvious choice was how he practiced mathematics.” One of Calabi’s main preoccupations, according to Kazdan, was to “ask interesting questions that no one else was thinking about.” The answers to those questions often had consequences of lasting significance.

Although Calabi made vital contributions to many areas of geometry, he is best known for his 1953 conjecture about a special class of manifolds. A manifold is a surface or space that can exist in any dimension, with an essential feature: A small “neighborhood” around every point on the surface looks flat. The Earth, for example, looks round (spherical) when viewed from afar, but a tiny patch of ground looks flat.

In graduate school at Princeton University, Calabi became interested in Kähler manifolds, named after the 20th-century German geometer Erich Kähler. Manifolds of this type are smooth, meaning that they have no sharp or jagged features, and they only come in even dimensions — 2, 4, 6 and up.

A sphere has constant curvature. Anywhere you go on the surface, regardless of the direction you set off in, your path bends the same amount. But in general, the curvature of manifolds can vary from one point to another. There are a few different ways that mathematicians measure curvature. One comparatively simple measure called the Ricci curvature was of great interest to Calabi. He proposed that Kähler manifolds could have zero Ricci curvature at every point even while satisfying two topological conditions that globally constrain their shape. Other geometers thought such shapes sounded too good to be true.

Shing-Tung Yau was initially among the doubters. He first came across the Calabi conjecture in 1970, when he was a graduate student at the University of California, Berkeley, and he was immediately transfixed. To prove that the conjecture was true, as Calabi had laid out the problem, one had to show that a solution to a very thorny equation could be found — even if the equation was not solved outright. That was still a big challenge because no one had ever solved an equation of this specific type before.

After spending a few years thinking about the problem, Yau announced at a 1973 geometry conference that he had found counterexamples that showed the conjecture was false. Calabi, who was at the conference, did not raise any objections at the time. A few months later, after giving the matter some thought, he asked Yau to clarify his argument. When Yau reviewed his calculations, he realized he’d made a mistake. The counterexamples did not hold up, suggesting that the conjecture might be correct after all.

Yau spent the next three years proving the existence of the class of manifolds Calabi had originally proposed. On Christmas Day in 1976, Yau met with Calabi and another mathematician, who confirmed the validity of his proof, establishing the mathematical existence of objects now called Calabi-Yau manifolds. In 1982, Yau won a Fields Medal, math’s highest honor, partly on the strength of this result.

Around that time, physicists trying to devise theories that unified the forces of nature started toying with the idea that fundamental particles such as electrons are in reality composed of exceedingly tiny vibrating strings. Different patterns of vibration manifest as different particles. For technical reasons, these vibrations only work out correctly in 10 dimensions.

Needless to say, the world does not appear to be 10-dimensional — there seem to be just three dimensions of space and one of time. By the mid-1980s, however, a group of physicists had realized that the six “extra” dimensions of the universe might be hidden in a minute Calabi-Yau manifold (less than 10^{−17} centimeters in diameter). String theory, as this physical framework was called, also held that the particles and forces of nature were dictated by the Calabi-Yau shape. This theory depended upon a property called supersymmetry, which arose from symmetry that was already built into a Kähler manifold — another reason why Calabi-Yau manifolds appeared to be the right fit for string theory.

By 1984, Yau already knew it was possible to construct at least 10,000 different six-dimensional Calabi-Yau shapes. It isn’t clear if our world is secretly filled with Calabi-Yau manifolds — concealed within dimensions far too small to be seen — but every year physicists and mathematicians publish thousands of papers probing their properties.

Yau said that the term comes up so often that he sometimes thinks his first name is Calabi. For his part, Calabi said in 2007, “I am flattered by all the attention that this idea has received,” owing to the connection with string theory. “But I’ve had nothing to do with that. When I first posed the conjecture, it had nothing to do with physics. It was strictly geometry.”

Calabi was not always determined to become a mathematician. His talent showed early — his father, a lawyer, quizzed him about prime numbers when he was a kid. But he decided to major in chemical engineering when he arrived at the Massachusetts Institute of Technology as a 16-year-old in 1939, after his family fled Italy at the outset of World War II. 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. He got his doctorate at Princeton and held a series of professorships before landing at Penn in 1964, where he would remain.

He never lost his enthusiasm for mathematics, continuing to carry out research well into his 90s. Chen, his former student, remembered how Calabi used to intercept him in the math department’s mailroom or in hallways: Their conversations could go on for hours, with Calabi scribbling down formulas on envelopes, napkins, paper towels or other scraps of paper.

Yau saved some of the napkins from his exchanges with Calabi. “I always learned from the formulas written on them, which conveyed Calabi’s uncanny sense of geometric intuition,” Yau said. “He was very generous about sharing his ideas and didn’t care about getting credit for them. He just thought that doing math was fun.”

Calabi called math his favorite hobby. “To follow your hobbies as a profession is the extraordinary luck I’ve had in my life.”