Invariants of singular spaces from noncommutative geometry

The common theme of our aims is to explain mysteries that cloud understanding of the geometry of surfaces and higher dimensional analogues (singular manifolds) including infinite dimensional ones. The surfaces are not smooth ones, they come from number theory, while the questions for higher dimensional situations are motivated by string theory and involve manifolds for which much is not known. We have new ideas on how to use analytic methods to study the singular geometries from number theory. For geometry coming from string theory there have been many new developments in the last two years, some of them due to us. The situation is ripe for substantial progress.