Title: Fenchel--Nielsen coordinates on Riemann surfaces and cluster algebras

Date: 10/15/2019

Time: 1:05 PM - 2:05 PM

Place: C304 Wells Hall

It is a 30(at least)-year old subject: it is known since long that both the standard Fenchel--NIelsen (lengths--twists) coordinates and (Y-)cluster coordinates (if we have holes) result in the same Goldman bracket on the set of geodesic functions on Riemann surfaces. The proof (of "local" nature in the first case and of "global" in the second) implies that these two sets of coordinates realise the same Poisson algebra. Nevertheless, constructing a direct transition between these two sets was elusive mainly due to complexity of the transition. For a sphere with 4 holes and torus with one hole, the corresponding formulas were obtained by Nekrasov, Rosly and Shatashvili in 2011. I present some preliminary results on the corresponding algebras in the general case and discuss possible relations to objects called Yang--Yang functionals.