Topics in the Spectral Theory of Random Operators and in Statistical Mechanics

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This research focusses on two groups of topics in mathematical physics: the spectral theory of random operators, and statistical mechanics models, with special focus on two and quasi-one-dimensional systems. A wide range of methods from mathematical physics, classical analysis, probability theory, and convex geometry will be applied to study some of the open problems in these areas. In addition, the connections between the two areas will be investigated. Several major problems pertaining to random band operators remain open after many years of research. Perturbative series (such as resolvent expansions) typically diverge in the physically interesting regimes, whereas the non-perturbative methods (supersymmetric formalism, transfer operators) have been only partially developed. The PI will develop perturbative and non-perturbative methods to improve the rigorous understanding of the spectral properties of random band operators. In two- and higher dimensional statistical mechanics, the PI will investigate the large scale properties of gradient models, as well as discrete models. A better understanding of the former will also shed light on the latter (for example, on two-dimensional height models). The PI will strive to combine the well-developed multi-scale and convexity methods with techniques from analysis and high-dimensional convex geometry. Broader impacts This project will develop the connections between mathematical physics and other fields of mathematics, especially, probability theory, classical analysis and convex geometry. These connections may lead to developments in all these fields. The PI has delivered numerous talks at seminars, colloquia, and international conferences in mathematical physics, analysis, probability, high-dimensional geometry, and the interactions between these areas.