The PI will continue his investigation of algebraic K-theory, cyclic and Hochschild homology of operator algebras and operator ideals; he will be studying constructions of invariants of operators (e.g., higher index invariants and regulators). He will also study exotic traces and resulting invariants; the applications to Classical as well as Fractal, and Noncommutative Geometry; ramifications for the theory of Dirichlet series and their behavior in the vicinity of the critical line of Dirichlet series. He also intends to investigate new phenomena in Noncommutative Geometry related to special derivations and exotic chain homotopy equivalences that replace Koszul resolution approach to de Rham theory, with a particular interest in possible applications to quantum groups, quantum homogeneous spaces and noncommutative manifolds.
Besides the areas mentioned in the previous paragraph, Algebraic and Analytic Geometry, Singularity Theory, Fractal Geometry and Mathematical Physics are the areas where the impact of the proposed investigations will be most significant.
