  This is a brief review of the main results of our paper arXiv:1101.1759 that
contains a complete global treatment of the compactified trigonometric
Ruijsenaars-Schneider system by quasi-Hamiltonian reduction. Confirming
previous conjectures of Gorsky and collaborators, we have rigorously
established the interpretation of the system in terms of flat SU(n) connections
on the one-holed torus and demonstrated that its self-duality symplectomorphism
represents the natural action of the standard mapping class generator S on the
phase space. The pertinent quasi-Hamiltonian reduced phase space turned out to
be symplectomorphic to the complex projective space equipped with a multiple of
the Fubini-Study symplectic form and two toric moment maps playing the roles of
particle-positions and action-variables that are exchanged by the duality map.
Open problems and possible directions for future work are also discussed.
