"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. Closely related, and in fact locally homeomorphic, are deformation spaces of locally homogeneous geometric structures. Such deformation spaces often arise as solutions to basic geometric problems, and their global properties provide powerful topological invariants, in particular for three- and four-dimensional manifolds.
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.
In recent years the deformation theory of geometric structures has received revived attention due to new developments, which involve in a deeper way the connections to Lie theory and gauge theory. Unexpectedly, many new examples of deformation spaces of geometric structures appeared. Two such developments are Higher Teichmueller theory and Anosov representations of hyperbolic groups, which generalize classical Teichmueller theory and the theory of quasi-Fuchsian representations to the context of Lie groups of higher rank.
The goal of the proposal is to understand the fine structure and internal geometry of deformation spaces of geometric structures, and to further develop the structure theory of discrete subgroups in higher rank Lie groups. Of particular interest are deformation spaces with appear in the connection with higher Teichmueller theory, because they are expected to be of similar mathematical significance as classical Teichmueller space."
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