An introduction to higher Teichmuller theory

2018/02/16 Fri 14:00 - 15:30
Room 402, RIMS
Ian Le
Perimeter Institute

Let S be a topological surface. Teichmuller space parameterizes the different ways of giving S the structure of a Riemann surface. Uniformization tells us that any Riemann surface can realized as a quotient of the upper-half-plane by a subgroup of PSL(2,R). Thus Teichmuller space to be viewed as a space of representations of the fundamental group of S into PSL(2,R). We will explain how cluster algebras and the theory of total positivity give an approach to Teichmuller theory which recovers classical ideas (like hyperbolic geometry, measured laminations, and quadratic differentials) while also permitting a generalization to PSL(n,R).