Speaker: Leonid Chekhov, Steklov Mathematical Institute

We identify the Teichmuller space $T_{g,s,n}$ of (decorated) Riemann
surfaces $\Sigma_{g,s,n}$ of genus $g$, with $s>0$ holes and $n>0$
bordered cusps located on boundaries of holes uniformized by Poincare with
the character variety of $SL(2,R)$-monodromy problem. The effective
combinatorial description uses the fat graph technique; observables are
geodesic functions of closed curves and $\lambda$-lengths of paths
starting and terminating at bordered cusps decorated by horocycles. Such
geometry stems from special 'chewing gum' moves corresponding to colliding
holes (or sides of the same hole) in a Riemann surface with holes. We
derive Poisson and quantum structures on sets of observables relating them
to quantum cluster algebras of Berenstein and Zelevinsky. A seed of the
corresponding quantum cluster algebra corresponds to the partition of
$\Sigma_{g,s,n}$ into ideal triangles, $\lambda$-lengths of their sides
are cluster variables constituting a seed of the algebra; their number
$6g-6+3s+2n$ (and, correspondingly, the seed dimension) coincides with the
dimension of $SL(2,R)$-character variety given by
$[SL(2,R)]^{2g+s+n-2}/\prod_{i=1}^n B_i$,
where $B_i$ are Borel subgroups associated with bordered cusps. I also discuss the
very recent results enabling constructing monodromy matrices of SL(2)-connections out of
the corresponding cluster variables.
The talk is based on the joint papers with with M.Mazzocco and V.Roubtsov