Historical development of Teichmüller theory

Abstract

Originally, the expression “Teichmuller theory” referred to the theory that Oswald Teichmuller developed on deformations and on moduli spaces of marked Riemann surfaces. This theory is not an isolated field in mathematics. At different stages of its development, it received strong impetuses from analysis, geometry, and algebraic topology, and it had a major impact on other fields, including low-dimensional topology, algebraic topology, hyperbolic geometry, geometric group theory, representations of discrete groups in Lie groups, symplectic geometry, topological quantum field theory, theoretical physics, and there are certainly others. Of course, the impacts on these various fields are not equally important, but in some cases (namely, low-dimensional topology, algebraic geometry, and physics) the impact was crucial. At the same time, Teichmuller theory established important connections between the fields mentioned. This, in part, is a consequence of the diversity and the richness of the structure that Teichmuller space itself carries. From a more subjective point of view, the result of pondering on these connections and applications demonstrates the unity of mathematics. The aim of this paper is to survey the origin of Teichmuller theory and the development of its early major ideas.