Complex Geometry and Lie Theory

In the late 1960s and early 1970s, Phillip Griffiths and his collaborators undertook a study of period mappings and variation of Hodge structure. The motivating problems, which centered on the understanding of algebraic varieties and the algebraic cycles on them, came from algebraic geometry. However, the techiques used were transcendental in nature, drawing heavily on both Lie theory and hermitian differential geometry. Promising approaches were formulated to fundamental questions in the theory of algebraic curves, moduli theory, and the deep interaction between Hodge theory and algebraic cyles. Rapid progress on many fronts was made in the 1970s and 1980s, including the discovery of important connections to other fields, including Nevanlinna theory, integrable systems, rational homotopy theory, harmonic mappings, intersection cohomology, and superstring theory.