Variational Methods in Mathematical Physics and Analysis

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The main focus of this project is the mathematical analysis of complex phenomena occurring in nature. The models under investigation originate from condensed matter physics and quantum information theory and describe various physical concepts ranging from quasiparticles in solids, to superconductivity and measures of entropy and entanglement. Besides understanding the mathematical and physical principles at work in some specific systems, analytical tools will be developed which are mathematically interesting beyond the context of these concrete problems and which will be applied to answer questions in pure mathematics. The project provides research training opportunities for graduate students and undergraduate students. More specifically, the PI will revisit the polaron problem, will investigate the connection between a microscopic and a macroscopic theory of superconductivity and study functional inequalities in both a commutative and a noncommutative setting. Attention will focus on a recently emerging area at the interface between harmonic analysis and calculus of variations. A unifying feature of all these problems is that from a certain (almost) optimality property one wants to conclude that the objects in question have a relatively simple structure. Mathematical tools to achieve this are often related to semiclassical analysis in a low regularity setting, and a major goal is to quantify the underlying non-commutativity, which is inherent to quantum physics.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.