CAREER: Randomness in Geometry and Dynamics

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This project involves using ideas from dynamical systems, probability theory and geometry to understand randomness in various geometric and dynamical contexts. The PI is developing new measures of complexity for parabolic dynamical systems using spacing statistics and gap distributions, and using renormalization methods to compute these invariants. The PI and his collaborators are also creating new notions of random geometric objects and structures, by placing appropriate measures on moduli spaces of geometric objects, and understanding properties of these random objects by computing moments of various functions with respect to these measures.The measures of complexity are being designed in this project are constructed to help detect new types of chaotic (`sensitive dependence on initial conditions') behavior of time evolution of physical systems in new situations, such as the motion of an electron on the Fermi surface of a metal in the presence of a magnetic field. Understanding the geometry and measure theory of moduli space of geometric objects is being used by the PI and a team of engineers in modeling and optimizing the performance of a lithium-ion battery. The research in this project develops and uses many computational experiments, as well as visualizations and animations of the models. Many of these are prepared by teams of students in the lllinois Geometry Lab (IGL), a facility founded by the PI, focusing on research, visualization and community engagement. The educational component of this award centers on development and expansion of the IGL, which involves a broad variety of participants: all the way from preschoolers to graduate students to teachers and educators. The activities of the IGL introduce them to mathematics as an exploratory endeavor, and develop a community of mathematical explorers.