We describe a construction which to a surface and a Iwahori-Hecke algebra associates an invariant which is a Laurent polynomial. More generally, this construction works for surfaces with boundary and behaves well under gluing, giving a non-commutative topological quantum field theory (TQFT). The invariant polynomial has surprising positivity properties, which are proven using Schur elements. We give some ideas on the combinatorial interpretation of the positivity linking to character varieties and integrable systems. Joint work with Vladimir Fock and Valdo Tatitscheff.
https://indico.math.cnrs.fr/event/7680/