Noncommutative Geometry and Index Theory

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In life, the order of doing things is extremely important. Doing job A before B may produce a completely different result from doing A after B. Noncommutative geometry studies this phenomena using ideas from geometry. While coordinate functions in classical geometry all commute, in the sense that the product AB equals BA, they do not commute in noncommutative geometry, where AB can be different from BA. In the last thirty years, noncommutative geometry has quickly grown into one of most active research areas in mathematics, with applications in many branches of mathematics and physics. In this project the principal investigator will study several interesting and important problems in noncommutative geometry and index theory, some of which are motivated by quantum field theory in physics.The principal investigator will pursue three research projects in noncommutative geometry and index theory. (1) He will study a long-standing index problem, raised by Alain Connes, about the index of a groupoid elliptic differential operator on a general Lie groupoid. (2) He will continue his program of using the joint force of noncommutative geometry and symplectic topology to study a duality conjecture, inspired from string theory in physics, about gerbes on orbifolds. (3) He will investigate an interesting index problem, motivated by multivariate operator theory, about some essentially normal Hilbert modules using the recent developments in complex analysis and noncommutative geometry.