Surfaces, triangles and Lie groups
The general subject of the talk will be to explain techniques helping in building surfaces in spaces. We will first recall how one can build (topologically) surfaces from triangles. Moving to geometry, and after a brief crash course in plane hyperbolic geometry, we will explain how one can describe « hyperbolic » surfaces out of « ideal triangles » in the hyperbolic plane, which themselves correspond to triple of points in the projective line.
After having explained these basic constructions, I will explain how one can move to describe triangles associated to Lie group — think of of the group of invertible matrices with complex coefficients –and explain how these constructions, will the help of ideas from Probability Theory, can help to solve our initial problem: building surfaces in spaces known as symmetric spaces.
The talk should be accessible to graduate students and I will assume no knowledge of hyperbolic geometry and Lie groups. This is motivated by a recent collaboration with J.Kahn and S. Mozes.
