New NSF Grant Aims To Learn More About Operator Algebras

Nate Brown, professor of mathematics and principal investigator on the grant, works on understanding operator algebras.

A new National Science Foundation (NSF) focused research group grant awarded to a mathematician in the Eberly College of Science aims to learn new information about operator algebras and their connections with dynamical systems.

Operator algebras were invented 100 years ago to be the framework for the field of quantum mechanics. In nature, there is a division between particles that exhibit quantum behavior and those that don’t. The line is drawn at the “Buckyball,” a 60-carbon molecule; smaller things exhibit wave-particle duality, larger ones don’t.

Now, a team of researchers led by Nate Brown, professor of mathematics and principal investigator on the grant, is trying to establish a similar line in operator algebras.

“Some algebras have finite noncommutative topological dimension and some do not. One of our primary goals is finding the analogue of the buckyball, the line separating algebras that fit into this beautiful dimension theory, and those that don’t,” said Brown.

Brown and his team, including co-principal investigators Guoliang Yu at Texas A&M University and Rufus Willett at the University of Hawaii, will also study connections with dynamical systems, a field of mathematics encoding the complexity of nature’s symmetries. The investigators will receive help from teams of researchers at the University of Münster and the University of Copenhagen, thanks to a cooperative agreement between the investigators and those universities.

Brown acknowledges that it could be many, many years before this work provides practical application.

“Advances in pure mathematics take a very long time to find application, and it’s often impossible to predict what those applications will be,” said Brown. “In the 17^th century, nobody dreamed of the internet. Yet every time we log in to a computer, our passwords are safely encrypted because of Fermat’s Little Theorem, a theoretical result proved 350 years ago. Similarly, the extraordinary accuracy of our GPS systems depends on non-Euclidean geometry, a 200-year-old piece of pure mathematics—until Einstein described space-time with it.”