In a first part, we shall recall basic facts about surfaces, their fundamental group and flat connections. Then, we shall explain various constructions of the moduli space of representations of surface groups in a Lie group G. We shall detail its smooth structure as well as its Poisson and symplectic structure. We will present a natural algebra of observables arising in this context, namely the Goldman algebra and the "spin networks" algebra. We will also explain how three-manifolds enter the picture.
We will then spend some time on Teichmüller Theory. We shall in particular explain the proof in recent work of Mirzakhani on recursion formulas for the volumes of Riemann moduli space, and how it relies on Mc Shane's identity. Coming back to the case where G is compact, we will also explain the complex structure of the moduli space. This will lead to the definition of conformal blocks: a natural finite dimensional vector space associated to this moduli space arising through a quantification procedure when G is compact. If time permits, some properties and constructions of conformal blocks will be shortly discussed: their dimension computed by Verlinde formula, the fact that they are acted upon by various grous like the mapping class group and loop groups.
Zeit: Di 10-12, Do 13-15
Ort: HG G 43 (Hermann-Weyl-Zimmer)
Beginn: 31. Oktober (Die Vorlesung findet nur jede zweite Woche statt)