Higher Teichmueller Theory

• Applicant: Professorin Dr. Anna Wienhard
• Subject Area: Mathematics
• Term: from 2013 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 237100532

Moduli spaces of flat bundles and representation varieties play a prominent role in various areas of mathematics. Historically such spaces first arose in the study of systems of analytic differential equations. Analytic continuation of their solutions led to monodromies and linear representations of the fundamental groups of the underlying manifolds, giving rise to the first examples of local systems. Closely related, and in fact locally homeomorphic, are deformation spaces of locally homogeneous geometric structures.Formally belonging to the realm of differential geometry and topology, the study of local systems and of deformation spaces of geometric structures heavily draws from Lie theory. Moreover, due to the ubiquity of these spaces, methods and viewpoints from various areas of mathematics such as dynamical systems, algebraic geometry, gauge theory, representation theory, partial differential equations, number theory and complex analysis can be combined, and their interplay gives rise to the richness of this subject. In recent year there has also been an increasing interaction with theoretical physics, which has been fruitful for both sides. Teichmüller space is the prototype of a space of local systems (or equivalently representations) which is globally homeomorphic to a deformation space of locally homogeneous geometric structures.It is a smooth cover of the moduli space of Riemann surfaces of fixed topological type and parametrizes (marked) conformal structures on a surface S. Teichmüller space is a central object in several areas of mathematics. It exhibits rich geometric structures. For example, Teichmüller space carries a natural complex structure and various mapping class group invariant metrics. It can be equipped with natural coordinates and admits several interesting ergodic flows. Teichmüller theory also played a central role in the recently completed geometrization program and the classification of hyperbolic three-manifolds.Quantizations of Teichmüller space lead to a geometric realization of a conformal field theory, the so called Liouville theory.Higher Teichmüller theory generalizes to Lie groups of higher rank, such as PSL(n), concepts, which in classical Teichmüller theory are related to the Lie group PSL(2). Higher Teichmüller theory emerged in the past twenty years and has developed into a very active field which also sparked interest form theoretical physics. The goal of the proposed research is to extend and further develop higher Teichmüller theory to a theory which fully generalizes the rich structure of classical Teichmüller space.

• DFG Programme: Research Grants