Operator K-theory and its applications to geometry and topology

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The investigator and his students plan to study the K-theory of operator algebras and its applications to geometry and topology of manifolds. 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 topology of manifolds. Examples of such applications include the existence problem for Riemannian metrics with positive scalar curvature and the Novikov conjecture on homotopy invariance of higher signatures. The investigator also intends to apply noncommutative geometry methods to study analysis on loop spaces of manifolds. The methods to be employed in this project include group actions on Banach spaces, coarse embedding into Banach spaces, geometry of expanders, and infinite dimensional index theory of elliptic operators.In classic 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 both 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 classic geometry and topology. Higher index theory serves as a bridge between noncommutative geometry and classic geometry and topology. The investigator and his students plan to apply noncommutative geometry methods to study problems in differential geometry and topology.