Noncommutative Geometry and Analytic Grothendieck Riemann Roch Theorem

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In classical geometry, functions on a geometric space, like the Cartesian coordinates on a plane, commute with each other. More precisely, the multiplication of coordinate functions does not depend on the order. However, the Heisenberg uncertainty principle in quantum mechanics indicates that the position and momentum, the two important observables of a quantum system, do not commute. Such noncommutative phenomena turn out to appear naturally not only in mathematics and physics but also everywhere in life. For example, clicking button A before B in a game may produce a completely different outcome from clicking B before A. Noncommutative geometry is an emerging branch of mathematics to study the geometry of "noncommutative spaces", where the functions, like the position and momentum, do not commute with each other. This project will explore various interesting noncommutative spaces motivated by problems in operator theory, singular spaces, and mathematical physics. The principal investigator proposes to investigate several projects in noncommutative geometry. In the first project, the principal investigator will take ideas from different branches of mathematics including algebraic geometry, differential geometry, and differential topology to investigate the Arveson-Douglas conjecture. An analytic generalization of the Grothendieck Riemann Roch theorem for (singular) analytic varieties will be developed to identify the K-homology class associated to the essentially normal Hilbert module in the Arveson-Douglas conjecture. A secondary invariant theory in noncommutative geometry will be introduced to understand the Hilbert modules associated to Brieskorn varieties. The aim of the second project is to develop a longitudinal index theory for open foliated manifolds. In particular, cyclic homology of a proper Lie groupoid will be applied as a key tool to study such an index theory. The third project is devoted to the study of the quantum geometry of étale gerbes on orbifolds. Inspired by results in physics, the principal investigator intends to establish a duality theory for the quantum geometry of étale gerbes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.