The study of spaces of complex structures on a Riemann surface, the so-called Moduli space and Teichmüller space is a classical and well-studied area of mathematics, with relations and interconnections with different areas of mathematics and also theoretical physics. In the case of surfaces with genus at least two, complex structures can be uniformized to hyperbolic structures, which are discrete, faithful representations of surface groups in the group of isometries of the hyperbolic plane. A natural generalization is to consider surface group representations in other semisimple Lie groups. In the last few years, spectacular advances have been made towards generalizing existing tools and techniques to the study of these representations, and their moduli spaces. Remarkably, in many cases there is a natural generalization of discrete, faithful representations which provides an analogue of Teichmüller space. Our program shall focus on two perspectives:The dynamic point of view, leadi...
