Two quantizations of the universal Teichmuller space

Quantization of the Teichmuller space of (punctured)
Riemann surface was done in late 90’s by Kashaev and independently by
Chekhov-Fock (later developed by Fock-Goncharov in the framework of
quantum cluster algebras). They both lead to certain projective
representations of the mapping class group of the surface. I’ll
discuss how the difference of these two approaches in the universal
case is captured in these projective representations. In the talk,
I’ll first explain the two different coordinate systems for
Teichmuller spaces used by the two approaches (which are related to
Penner’s lambda lengths), and the relationship between them. Then I’ll
discuss what it means to quantize, and how to get quantization of
Teichmuller space. Finally I’ll say a few words about how this leads
to (two different) projective representations of mapping class group.
I’ll make the talk as elementary as possible, and will assume only
very basic knowledge on geometry.