Noncommutative triangulated surfaces and the Laurent phenomenon

The celebrated Ptolemy relation plays an important role in various studies of triangulated surfaces including hyperbolic geometry, geometrical applications of cluster algebras and so on. We will discuss a noncommutative "categorification" of the relation which is based on the theory of noncommutative Plücker coordinates developed by I. Gelfand and the speaker. The noncommutative Prolemy relations provide several examples of the noncommutative Laurent phenomenon and lead to new noncommutative topological invariants of the surfaces. (Joint work with Arkady Berenstein from University of Oregon)