Alladi Ramakrishnan Hall

Old and New on the Moduli Spaces of Local Systems on Surfaces

Rishi Raj

Dept. of Mathematics, Yale University

A surprising discovery of the 1980's was the relationship between
knot invariants and quantum groups. Witten gave an elegant explanation of
this relationship, by deriving it from the 'quantization' of moduli spaces
of local systems on surfaces. I will give an elementary account of this
derivation, illustrating it with the Jones polynomial. This is the "old"
part of the story.

More recently, new structures were discovered on the same moduli spaces,
inspired by Teichmuller theory, and the study of canonical bases in
representation theory. These new structures give an a priori different way
of quantizing the moduli spaces. I will explain that the old and new
quantizations are essentially equivalent. As a corollary, we prove the
'Modular Functor conjecture' of Fock-Goncharov.

If time remains, I will also state a result relating all of this to the
'categorification' of cluster algebras.