K-Theory of Operator Algebras and Its Applications
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abstract

In classical geometry, one studies geometric objects whose coordinates commute. Noncommutative geometry is a mathematical theory specifically designed to study "geometric objects" whose coordinates do not commute but which do occur naturally in mathematics and physics. In the last decade or so, with the help of new ideas from noncommutative geometry, great advances have been made towards the solutions of long-standing problems in classical geometry and topology. K-theory, higher index theory, and secondary index theory serve as bridges between noncommutative geometry and classical geometry and topology. The principal investigator and his students plan to apply noncommutative geometry methods to study problems in differential geometry and topology.The K-groups of certain operator algebras are receptacles of higher indices and secondary index invariants of elliptic differential operators and have important applications to problems in differential geometry and in the topology of manifolds. Examples of such applications include estimation of the size of the moduli space of all Riemannian metrics with positive scalar curvature and questions concerning the rigidity or nonrigidity of a manifold. The principal investigator and his students plan to apply the techniques of quantitative operator K-theory and dynamic complexity to study K-theory of operator algebras, higher index theory, and secondary index theory. The principal investigator and his students also intend to apply quantitative techniques to study the isomorphism conjectures in algebraic K-theory.