Chern

Chern is a surname that most commonly refers to the Chinese-American mathematician Shiing-Shen Chern (1911–2004), a foundational figure in differential geometry. His work helped define and organize modern approaches to the geometry of differentiable manifolds and the study of curvature.

In mathematics, several central concepts bear the name Chern, named after Shiing-Shen Chern. Chern classes are characteristic classes associated with complex vector bundles, providing topological invariants that live in cohomology. They, together with other characteristic classes, classify bundles and relate to geometric properties. The total Chern class encodes all individual Chern classes, and the first Chern class plays a basic role for line bundles.

Chern-Weil theory gives a differential-geometric construction of these classes using curvature forms from connections on bundles, producing de Rham cohomology representatives that remain invariant under changes of connection. The Chern character is a related construction that maps K-theory to cohomology, translating bundle data into a systematic cohomological invariant.

Chern-Simons theory, developed in collaboration with James Simons, is a three-dimensional topological quantum field theory that assigns invariants to three-manifolds and gauge fields. It has influenced both mathematics and theoretical physics, including developments in geometry, topology, and quantum field theory.

Chern’s publications and ideas helped shape differential geometry into a rigorous and influential field, with concepts such as Chern classes, Chern-Weil theory, Chern character, and Chern-Simons theory remaining central in geometry, topology, and related areas. The term Chern in this context generally denotes the legacy of Shiing-Shen Chern and the mathematical structures named in his honor.