This project is concerned with applications of noncommutative geometry to problems in geometry and analysis. The main emphasis is on the index problems and related question solutions of which require extensive use of tools of cyclic cohomology and K-theory. Such problems arise naturally in the study of the index theory for foliations. Another important aspect of this work is the application of the formal deformation techniques. This requires further development of the theory of formal deformations.
The proposed research belongs to the area of noncommutative geometry. The starting point of the noncommutative geometry is the appearance in physics and mathematics of the objects which, while having clearly geometric nature, can not be adequately described by the means of the classical geometry. Noncommutative geometry is based on the fundamental observation of A. Connes that for an extraordinarily wide class of such nonstandard spaces, one can naturally associate an algebra that fails to be commutative, but behaves well in every other respect. In noncommutative geometry such an algebra, called a noncommutative space, is the central object of study. The present project is concerned with applications of the powerful techniques developed within the noncommutative geometry to the study of the classical spaces.
