Variational Questions, Stability, and Dynamics

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The search for a better understanding of many physical processes, and the evolution equations that govern them, is in part a quest for new and more precise mathematical inequalities. The answers to such questions as "How fast can this process go?" and "How rapidly can information be communicated through this channel?" often turn on the discovery of new mathematical inequalities, which are the central theme of much of the project. The problems to be investigated are of significant interest within pure mathematics as well. But because they are motivated by problems arising in other fields, especially physics and quantum information theory, their solution will have impact and applications outside of pure mathematics. Students will be involved in the project, contributing to training of the next generation of researchers.The connection between mathematical inequalities and physical processes runs both ways. So, in trying to prove a particular inequality, one may try to relate it to a simple and well understood evolution equation. This area of research has been fruitful not only in producing results that are of interest to a wider scientific community, but also in engaging the interest of Ph.D. students. The project places particular emphasis on proving functional inequalities in sharp form. This means to completely solve the following variational problem: find the minimum value of some functional and determine the full set of equality cases. Some of the questions go further and seek stability results for such sharp inequalities, i.e., theorems that assert that when a certain function almost yields equality in a functional inequality, then that function is close in some metric to one of the cases of equality. Such stability results have many important implications. Both their proofs and applications will be worked on by the PI and his collaborators. This research will produce not only significant new mathematics, but results that are relevant to the physical sciences and even engineering as well.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.