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Deformation Spaces of Geometric Structures

Final Report Summary - GEOMETRICSTRUCTURES (Deformation Spaces of Geometric Structures)

In the 19th century the German mathematician Felix Klein introduced a new concept of geometry, instead of looking at geometric quantities that are measured (lengths, angles etc) he put the focus on the transformations that leave these quantities invariant. This approach had a tremendous influence and governs the modern view of geometry. A geometry is determined by a Lie group G acting on a space X.
The ERC project "Deformation spaces of geometric structures" studied the interplay between actions discrete subgroups of G on the one hand and the question if a given manifold M can be endowed with a geometric structure so that a small patch in M looks exactly like a small patch in X. A special focus of the project was on a new mathematical research area called higher Teichmüller theory and Anosov representations, which emerged in the past 20 years. In particular, the project provided a new underlying framework for higher Teichmüller theory, which is based on a generalization of the notion of total positivity in Lie groups, which plays an important role in several areas in mathematics, from representations theory to statistics. The project established several new geometric and dynamical properties of Anosov representations. The outcomes of the project open up several new directions of research that will be pursued in future years.