Commutative Algebraic Methods in Multivariable Operator Theory and Applications

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The objective of this proposal is develop a systematic theory of commutative algebraic invariants in operator theory, centering around the dimension invariant and the Samuel multiplicity of a tuple of commuting operators acting on a Hilbert space. These algebraic invariants have found applications to purely analytic problems in operator theory including the study of Hilbert spaces of analytic functions, and Fredholm theory. Further connection with Nevanllina-Pick kernels will be explored in the proposed research.Hilbert space operators can be viewed as quantization of classical objects such as holomorphic functions. Operator theory provides an infinite dimensional approach to problems in a finite dimensional space. The subject originates from providing a mathematical foundation to Quantum Mechanics, and has found applications in quality control in Electrical Engineering. The proposed research will introduce new techniques from commutative algebra, another area of mathematics, into operator theory, hence not only enriching the subject, but also enhancing the ties between two subjects in mathematics.