Shiing-Shen Chern

Shiing-Shen Chern the Chinese-American mathematician (1911-2004) regarded as one of the greatest geometers of the twentieth century, whose work on differential geometry, characteristic classes and fibre bundles transformed the subject and whose Chern classes are fundamental tools throughout modern mathematics and physics.

Early Life and Education

Shiing-Shen Chern was born on 28 October 1911 in Jiaxing, Zhejiang province, the son of a scholar and official. The family moved to Tianjin, where he attended a modern school and displayed a marked talent for mathematics. He entered Nankai University in 1926 and graduated in mathematics in 1930.

He continued his studies at Tsinghua University, where he worked under the geometer Sun Guangyuan and completed a master's thesis on projective differential geometry. In 1934 he went to Europe on a fellowship, studying first at the University of Hamburg under Wilhelm Blaschke and then in Paris, at the invitation of Elie Cartan.

Cartan, one of the deepest mathematicians of the age, introduced Chern to the method of moving frames and to the theory of exterior differential forms, the techniques that would become the instruments of his greatest discoveries. The encounter was decisive, and Chern later described Cartan as the greatest influence on his mathematical thinking.

Differential Geometry and the Moving Frame

Chern returned to China in 1937 and taught during the war years at the universities that had been relocated to the southwest. Working in difficult conditions he continued his research, producing a series of papers on differential geometry, on the geometry of submanifolds and on the theory of surfaces embedded in higher-dimensional spaces.

His mastery of Cartan's methods allowed him to solve problems that had resisted other approaches, and he gradually developed his own distinctive style, combining global reasoning with explicit local computation. He became known for results on the geometry of webs, on minimal surfaces and on the intrinsic invariants of manifolds.

After the war he went to the United States, spending time at the Institute for Advanced Study and then joining the faculty of the University of Chicago in 1949. There, in close contact with a remarkable group of topologists and geometers, he entered the most productive period of his career.

Characteristic Classes and Global Geometry

Chern's most celebrated achievement was the theory of characteristic classes, invariants that measure the twisting of a fibre bundle over a curved space. In a series of papers in the 1940s he used the method of differential forms to construct these classes explicitly and to prove that they are invariant under change of connection, a result known as the Chern-Weil theory.

These quantities, now called Chern classes and Chern numbers, became indispensable tools. They connect the local curvature of a space to its global topology, and they appear throughout geometry, topology, algebraic geometry and theoretical physics. Their influence on the classification of manifolds and on the theory of index theorems has been profound.

Chern also proved a famous theorem on the topology of complex manifolds, showing that the Chern classes of a class of manifolds satisfy certain restrictions, and he established the intrinsic nature of the characteristic classes connected with the structure of the tangent bundle. These results opened entirely new directions in global differential geometry.

Later Career and Influence

Chern spent many years at the University of California, Berkeley, where he built one of the world's leading centres of geometry. He supervised dozens of doctoral students, many of whom became prominent mathematicians, and his seminars and lectures attracted scholars from around the world.

He worked on the geometry of the Chern-Simons theory, developed with James Simons, which provided a new invariant of three-dimensional manifolds and later found remarkable application in theoretical physics. He also studied holomorphic curves, integral geometry and the geometry of surfaces, continuing to publish important work late into his life.

He received many honours, including the National Medal of Science, the Wolf Prize and the Shaw Prize, and he was elected to the National Academy of Sciences and to a great number of foreign academies. He was especially honoured for his role in bridging Chinese and Western mathematics.

The Nankai Institute and Legacy

In his later years Chern devoted himself to the development of mathematics in China. He helped to found and directed the Nankai Institute of Mathematics at Nankai University, his alma mater, and he supported conferences, exchanges and programmes that brought international mathematics to his homeland and young Chinese mathematicians to the world.

He served as the director of the Mathematical Sciences Research Institute at Berkeley and as a guiding figure in several international bodies. Through his personal generosity and his example he encouraged a generation of Chinese and Chinese-American mathematicians, and he took particular pleasure in the growing strength of the subject in East Asia.

Chern died on 3 December 2004 in Tianjin at the age of ninety-three, in the city associated with his youth and with the institute he had helped to create. His name lives on in the Chern classes and in the many theorems that bear it, and he is widely regarded as the founder of a whole school of modern differential geometry.

See also

• Chern class
• Differential geometry
• Chern-Simons theory
• Nankai University