It is a fundamental scientific and mathematical problem to understand how microscopic interactions influence physical phenomena at a macroscopic level. This project investigates this question for a class of mathematical models of systems that arise in several scientific disciplines. Applications range from game theory, which includes decision strategies, investors' behavior, and socioeconomic sciences, to astrophysics and plasma physics. A fundamental component of this project concerns the contribution to mathematical education and training of undergraduate and graduate students in STEM fields.
The principal investigator will study a class of partial differential equations where microscopic effects are described by certain non-local operators. This research contains two different projects. The first project concerns the study of global well-posedness of non-local kinetic equations, with particular attention to the Landau equation. The Landau equation is a well-known equation that is used to model plasma with predominant grazing-type collisions. At present the well-posedness of the Landau equation for degenerate potentials is still an open problem and has given rise to several unresolved conjectures. The second project is related to a class of free boundary problems. The theoretical understanding of these and related projects presents analytical challenges and requires mathematical techniques that come from analysis, probability, and geometry. The skills developed through participation in this research project will be taken by students into the workforce.
