Operator Analysis and Applications

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Operator Theory was originally developed as a mathematical language to describe Quantum Mechanics. It has spread to many other areas in pure and applied mathematics, such as Control Theory. Control Theory is the design of systems like automatic pilots and self-driving cars that must react, but not over-react, to their environment. As the systems get more complex, and more features are incorporated, new problems arise. The PI will work on developing new mathematical tools to deal with these problems. This project also contributes to US workforce development through the training of graduate students.The PI will study problems in Operator theory, function theory, and the interaction between them. Non-commutative functions are functions whose input is two (or more) operators, and whose output is an operator. This is a relatively new field, but already has applications in control theory, algebraic geometry, and semi-definite programming. The PI will study non-commutative functions, which can often shed light on the commutative case, and have recently become important in Control Theory. The methods will be a combination of techniques from multivariable operator theory, functional analysis and several complex variables. In addition, the PI will work on using mathematical models to study the development of Alzheimer's disease, the analysis of acoustic signals, and risk measurement models for COVID-19.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.