Noncommutative Geometry

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The proposal involves investigations in the areas of algebraic K-theory, Homological Algebra, with an emphasis on Cyclic and Hochschild Homology, and Infinitesimal Geometry. Specifically, invariants of operators are being constructed and studied: exotic traces, higher index invariants and regulators, as well as their ramifications for the theory of Dirichlet series and their behavior in the vicinity of the critical line. The structure, homology, and invariants associated to the algebras of differential operators and symbols are the focus of the investigation in Infinitesimal Geometry. Finally, new phenomena in Noncommutative Geometry related to special derivations and exotic chain homotopy equivalences that replace Koszul resolution approach to de Rham theory are the subject of the investigation in Homological Algebra. The projects in this proposal are concerned with fundamental topics in Noncommutative Geometry, Homological Algebra, Functional and Global Analysis. They offer novel approaches to some classical subjects in other areas of Mathematics and Mathematical Physics, in particular, Algebraic Geometry -- including its Arithmetic aspects, Singularity Theory, Fractal Geometry, Analysis on Quantum Manifolds.