CAREER: Renormalization and higher rank parabolic actions

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). This project aims to advance knowledge on parabolic systems, which refer to a class of complex systems that appear in many areas of mathematics as well as physics. There is a significant educational component: this project plans to support the activities of the Laboratory of Experimental Mathematics, of which the PI is a co-director, as well as the Girls Talk Math (GTM) program, both hosted at the University of Maryland. The former activity is aimed at engaging undergraduate students in research, and the latter at attracting high-school students from underrepresented groups towards the STEM-disciplines.Parabolic systems are the types of dynamical systems that are neither chaotic (that is, exhibit exponential complexity), nor completely rigid. Parabolic systems typically exhibit a type of polynomial complexity. An example of one such system is the motion of a particle on a frictionless and pocketless billiard table of non-rectangular polygonal shape with completely elastic collisions at the boundary. Higher rank parabolic systems appear in the study of aperiodic tilings, closely connected to applications to mathematical physics. Parabolic systems can be studied through techniques known as renormalization methods. A significant component of the project is devoted to developing renormalization tools in the language of operator algebras, that is, using invariants coming from operator algebras to obtain dynamical invariants.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.