Regularity Properties and K-Theory of Crossed Product Operator Algebras

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Algebra, analysis, geometry, and dynamics are some of the major branches of modern mathematics. Algebra studies the rules to work with mathematical symbols in ways that generalize addition and multiplication. Analysis deals with approximations of numbers, functions and other mathematical objects. Geometry concerns itself with notions such as shape, distance, angle, etc. Dynamics studies motions and in particular their long-term behaviors. This research project lies in the area of operator algebras and noncommutative geometry, which uses advanced techniques from algebra and analysis to study mathematical problems that are often geometrically and dynamically motivated. Outreach activities to foster communication and collaboration, especially among early-stage mathematicians will also be carried out.This project focuses on regularity properties and K-theory of operator algebras of dynamical nature, in particular, crossed product C*-algebras. It is motivated by applications in several areas of mathematical research. On the one hand, to further advance the (already spectacular) applications of noncommutative geometry to the celebrated Novikov conjecture in differential topology, the PI aims to study the K-theory of group C*-algebras of groups of diffeomorphisms on smooth manifolds. On the other hand, the classification program for simple separable nuclear C*-algebras has spawned and highlighted a number of regularity properties of C*-algebras, and the PI is working on transporting these properties to the dynamical setting, so as to aid future applications to the classification and structure theory of topological dynamical systems.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.