New Developments at the Interface of Banach Algebras and Complex Analysis

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This project will advance research at the interface of complex analysis and functional analysis, areas of mathematics that provide the mathematical tools used in physics, engineering, and computational chemistry. Several longstanding problems will be studied using new concepts recently introduced by the principal investigator. The results of this research project will be disseminated via talks in seminars and conferences and via publications in widely available journals. The principal investigator will continue to mentor junior mathematicians in related areas by suggesting to them problems motivated by this research project that are well matched to their particular strengths. He will also continue to encourage students to pursue university degrees in STEM fields. The principal investigator will use methods from functional analysis and complex analysis in one and several variables to study problems in four distinct, related areas of mathematics: the corona problem in several complex variables, Arveson's conjecture in multivariable operator theory, problems concerning analytic structure in commutative Banach algebras, and problems concerning homeomorphism groups. A major focus of the research will be the close connection between commutative Banach algebras and analyticity which, despite over 60 years of investigation by many mathematicians, remains in many ways mysterious. The principal investigator recently discovered a new concept of analytic set: in addition to providing a new perspective on analytic structure in commutative Banach algebras, this concept also provides a new perspective on the corona problem and suggests the development of a field that could be described as "nonsmooth complex analysis". Another major focus of the research will be Banach algebras invariant under group actions and their connections to an important conjecture of Arveson concerning Hilbert modules with wide ranging ramifications in operator theory and C*-algebras.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.