Quantitative Operator K-theory and Applications

In classical geometry, one studies geometric objects whose coordinates commute. Noncommutative geometry is a mathematical theory specifically designed to study "geometric objects" whose coordinates do not commute but which do occur naturally in mathematics and physics. In the last decade or so, with the help of new ideas from noncommutative geometry, great advances have been made toward the solutions of long-standing problems in classical geometry, topology, and mathematical physics. K-theory serves as a bridge between noncommutative geometry, classical geometry, topology, and mathematical physics. The principal investigator and his students plan to develop a quantitative operator K-theory to investigate problems arising from these areas of mathematics. This project will contribute to US workforce development through mentoring of graduate and post-doctoral students. The principal investigator and his students will focus on the applications of quantitative operator K-theory to problems in geometry, topology, almost representation theory of discrete groups, and structure theory of operator algebras. In particular, the investigator intends to apply quantitative operator K-theory techniques to solve problems in higher index theory of differential operators, quantitative behavior of scalar curvature, norm stability of almost representations, and the universal coefficient problem in structure theory of 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.