Principal Investigator: Anna Wienhard
The goal of the proposed project is to further develop the emerging field of higher Teichmueller theory. Higher Teichmueller spaces are connected components of the variety of representations of the fundamental group of a surface into semisimple Lie groups, which share essential properties with Teichmueller space. Classical Teichmueller space plays an important role in various fields of mathematics due to its rich structure and the fact that it is a smooth cover of the moduli space of Riemann surfaces. For higher Teichmueller spaces many potentially interesting structures are yet to be discovered and the relation to the moduli space of Riemann surfaces still needs to be clarified. The proposed research focusses on identifying the geometric objects (similar to hyperbolic surfaces in the case of classical Teichmueller space) which are parametrized by higher Teichmueller spaces. These objects will then be used to define algebraic, geometric and dynamical structures on higher Teichmueller spaces, and we expect that they will help to construct a natural map from higher Teichmueller spaces to classical Teichmueller space.
Teichmueller space is the space of all geometric surfaces with a fixed combinatorial type. Because it parametrizes all surfaces, Teichmueller space is a fundamental mathematical object which also plays an important role in theoretical physics. Every point in Teichmueller space corresponds to a tiling of the Euclidean plane, the two-dimensional sphere or in most cases the hyperbolic plane by one polygon and its images under subsequent reflections in the sides. (Nice pictures inspired by such tilings appear in M.C. Escher's work.) The combinatorial type of the surface is determined by the number of vertices of the polygon and the gluing data, but the geometry of the surface depends on the length of the sides and the angles at the vertices of the polygon. Teichmueller space is the example of a moduli space of a pattern (the tiling) with rich symmetries (the reflections in the side of the polygon). Such moduli spaces can be efficiently studied via the group generated by the symmetries of the pattern. Higher Teichmueller spaces are moduli spaces of more complicated patterns in higher dimensional spaces. However, the symmetry group of these patterns is the same as the symmetry group of the tilings associated to classical Teichmueller space. Therefore one expects higher Teichmueller spaces to parametrize geometric objects which are not surfaces themselves, but closely related to surfaces. The proposed research focusses on identifying these geometric objects and studying their finer structure.
The project proposes various activities to involve graduate students in research and teaching, as for instance a yearly mathematical retreat for graduate students to study, explore and present a mathematical topic in a group, and small workshops with mini-courses which allow graduate students to get insight into the working of modern research collaborations.