On 3 December 2004, Shiing-Shen Chern died in Tianjin, China, of a heart attack, and the mathematics community lost one of its legendary greats. Chern’s pioneering ideas influenced not only his own field of differential geometry, but also many parts of mathematics and theoretical physics. He played a crucial role in the founding of mathematics institutes in both the US and China. As a teacher, colleague, and friend, he was revered for his warmth and gentle nature.
Chern was born in China’s Zhejiang province on 28 October 1911. His undergraduate education was at Nankai University in Tianjin. In 1936 he completed his doctoral work with Wilhelm Blaschke at the University of Hamburg in Germany. Chern spent a year in Paris studying with Élie Cartan before returning to China to assume a professorship at Tsing Hua University. He visited the Institute for Advanced Study in Princeton, New Jersey, in 1943–45, and it was then that he began to produce his profound work on characteristic classes and curvature. After his return to China in the late 1940s, he helped establish a mathematics institute at the Academica Sinica in Beijing. He moved to the United States in 1949 to assume a post at the University of Chicago and in 1960 moved to the University of California, Berkeley. His originality in differential geometry and his lectures on that subject were legendary, as was his mentoring of young mathematicians, many of whom are now leading researchers in geometry.
Upon his retirement in 1979, he helped found the Mathematical Sciences Research Institute in Berkeley, California, and served as its first director (1981–1984). In 1985, he founded and directed the Nankai Institute of Mathematics in Tianjin. In 2000, following the death of his wife, Chern moved from Berkeley to Tianjin, where he remained until his death.
A major theme of Chern’s mathematical work was the relationship between curvature and topology. Consider a curved surface, such as a sphere or a torus, sitting in space. In the first part of the 19th century, following the work of earlier geometers, Carl Friedrich Gauss introduced the curvature that bears his name and proved the Gauss—Bonnet theorem: The total curvature of a surface—that is, the integral of the curvature over the entire surface—is determined by its topology, specifically its Euler characteristic. (The Euler characteristic of a surface is V − E + F, where in any triangulation V is the number of vertices, E is the number of edges, and F the number of faces. It is 2 for the sphere and 0 for the torus.)
Differential geometry in higher dimensions was developed in the second half of the 19th century through the work of Bernhard Riemann and his successors. In the early 20th century, Albert Einstein’s general relativity gave a big boost to the study of differential geometry. Yet not until 1943, in a paper Chern considered to be his best work, was the link between curvature and topology established intrinsically in higher dimensions. André Weil and Carl Allendoerfer had just proved a formula expressing the Euler characteristic of a manifold in terms of curvature, but their proof is for submanifolds of Euclidean space. Chern’s proof is intrinsic—that is, it applies to abstract manifolds and uses only the geometrically relevant information. His intrinsic point of view led him to introduce new techniques in the theory of fiber bundles, techniques that had a profound effect on geometry and physics in the latter part of the 20th century.
Imagine a manifold as a model of spacetime. In Einstein’s theory the manifold is 4-dimensional, in superstring and M-theory it is 10- or 11-dimensional, and for mathematical purposes it can have any number of dimensions. A fiber bundle over the manifold attaches to each point of spacetime some internal space. In statistical mechanical models, for example, one might imagine a spin taking values in this internal space, which is typically a sphere or a finite set of points. In particle physics the internal space represents the quantum state of a particle, such as a photon or neutron, in which case the space is linear. As one moves around spacetime, the internal space may twist, and the theory of characteristic classes was developed by Chern and others in the mid-1940s to measure the twisting. As with the formula for the Euler characteristic, there are close ties to curvature. There are also important “secondary” invariants—the Chern—Simons invariants, introduced by Chern and James Simons in 1971—which appear in a differential geometric context. All of these mathematical ideas enter profoundly into condensed matter physics (Berry’s phase), quantum field theory, and superstring theory—indeed, the ideas are embedded into our modern notions of geometry.
Chern was honored with many prizes, including the National Medal of Science (1975); the Wolf Prize (1983); and the first Shaw Prize (2004), which cited Chern’s “initiation of the field of global differential geometry and his continued leadership of the field, resulting in beautiful developments that are at the centre of contemporary mathematics, with deep connections … to all major branches of mathematics of the last sixty years.”
