Noncommutative Geometry: Its Applications to Geometry and Analysis

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This project will apply noncommutative geometry to study several problems in geometry and analysis. The project will study index theory on various singular spaces. Using the recent developments in Molino's theory, Hopf cyclic theory, and algebraic local index theory, PI aims to obtain an explicit description of the topological index of a transverse elliptic differential operator of a riemannian foliation. PI will also study a covering index theorem on orbifolds and investigate homotopy invariance properties of signature numbers on orbifolds. The project will develop some noncommutative geometry tools, especially groupoid Mackey machine, to study a physics conjecture about duality between gerbes on orbifolds. Finally, several interesting applications of Connes-Moscovici's Hopf algebras will be made to geometry, analysis, and number theory.The project is an interplay between analysis and geometry.Noncommutative geometry tools will be used to study geometry and topology of spaces with singularities; and ideas and motivations in geometry will lead to new structures and developments in operator algebras and harmonic analysis.