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Teichmüller theory is rich in applications to topology and physics. By way of the mapping class group the subject is closely related to knot theory and three-manifolds. From the uniformization theorem, Teichmüller theory is part of the more general study of Kleinian groups and character varieties. Conformal field theory and quantum cohomology make use of the algebraic and geometric properties of the Riemann moduli space.
The goal of this exposition is to present some of the more recent activity using analysis, and in particular harmonic maps, in the context of Teichmüller theory, representations of surface groups, mapping class groups, and Weil–Petersson geometry. Topics have been selected in order to illustrate the theme that the analytic and topological points of view complement each other in a useful way. For example, we shall present four different proofs of the fact that Teichmüller space is a cell, and we shall discuss the recent completion of a harmonic maps approach to Teichmüller’s existence and uniqueness theorems on extremal quasiconformal maps.