K-theory of Operator Algebras and Its Applications to Geometry and Topology

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In classical geometry, one studies geometric objects whose coordinates commute. Noncommutative geometry is a mathematical theory specifically designed to handle "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, there have been a series of advances towards the solutions of long-standing problems in classical geometry and topology. Higher index theory serves as a bridge 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 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 intends to apply the techniques of controlled operator K-theory, dynamic complexity, and finite embeddability into Banach spaces to study K-theory of operator algebras and higher index theory. He also intends to apply noncommutative geometry methods to study analysis on infinite-dimensional spaces (e.g., loop spaces of manifolds).