Arveson-Douglas Conjecture and Its Applications

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This project focuses on an area of research called the Arveson-Douglas Conjecture, which belongs to the general field of operator theory. It embodies many challenges and new, exciting mathematics. The conjecture was originally made around 2000, and a large body of literature has been accumulated on the subject while many unsolved problems remain. It has connections with and applications to many other parts of mathematics; for example, it is connected to index theory, which attracts the attention of many researchers from various fields. Another connection is to the holomorphic extension problem in several complex variables, another important branch of mathematical analysis. The extension problem itself goes back to the 1970s, and recent progress on the Arveson-Douglas Conjecture sheds new light on this old problem. An important part of any research in mathematics is the search for new tools and methods, and the principal investigator will take concrete steps toward that goal. On the technical side, the Arveson-Douglas Conjecture makes strong connections with geometry and several complex variables. The concrete steps in this research include: the study of a recently defined property - the asymptotic stable division property - for submodules; a generalized version of the holomorphic extension problem; identifying index elements and index formulas on modules; extending known results to strongly pseudoconvex domains. The research is aimed at building machinery that allows one to treat global properties of complex analytic sets, especially algebraic sets, through local analysis. New techniques involving tools from operator theory, harmonic analysis, several complex variables and topology will be developed to treat these problems. The principal investigator will also study Toeplitz operators and Toeplitz algebra on strongly pseudoconvex domains.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.