John E. McCarthy, the Spencer T. Olin Professor of Mathematics and chair of the Department of Mathematics and Statistics in Arts & Sciences at Washington University in St. Louis, received a five-year $450,000 grant from the National Science Foundation for research on operator analysis and applications.
Operator theory was originally developed as a mathematical language to describe quantum mechanics and has spread to many other areas in pure and applied mathematics. One such area is control theory, which is used in the design of systems such as automatic pilots and self-driving cars that must react — but not overreact — to their environment.
As the systems get more complex, and more features are incorporated, new problems arise. As principal investigator for the grant, McCarthy will work on developing new mathematical tools to deal with these problems.
McCarthy will study problems in operator theory and 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. McCarthy will study non-commutative functions, which often can 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, McCarthy 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.