Research in Analysis

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This research project covers a range of problems central to classical analysis and partial differential equations. The research will focus on three topics: approximation theory, nonlinear nonlocal evolution equations, and the spectral theory of Schrodinger operators. In approximation theory, the research concerns problems related to estimating the size of orthonormal polynomials. Understanding the asymptotics of the orthonormal polynomials has proved to be crucial for the progress in several fields, including mathematical physics and probability. The second main theme of the research is analysis of two-dimensional active scalar equations. These nonlinear and nonlocal evolution equations appear in the fluid dynamics and atmospheric and ocean sciences. The physical processes described by these equations are known to be highly unstable, and this project aims to develop a mathematical explanation for that phenomenon. The third topic of this project is spectral theory of Schrodinger operators. The Schrodinger equation is one of the basic equations in mathematical physics. The project will focus on understanding how the properties of the potential function affect the scattering of waves. In the area of approximation theory, the research investigates the following classical question: what is the size of the polynomial orthonormal on the unit circle (or the real line) with respect to a measure from a given class. Various ways to define the size and various classes of measures will be considered. For example, the project will study a class of weights that have a fixed oscillation around a given constant. The second part of the project will mainly focus on two active scalar equations: the two-dimensional Euler equation of fluid dynamics and the surface quasi-geostrophic equation. Questions of instability will be addressed; in particular, patch dynamics will be studied and the merging mechanism of a centrally symmetric pair will be analyzed. The third topic of this project will address multidimensional scattering. The Schrodinger evolution with rough potential will be considered and the classical questions of the scattering theory will be studied. To carry out the proposed research, methods of harmonic and complex analysis will be used along with techniques from nonlinear analysis and the theory of partial differential equations. Through the training of graduate and undergraduate students the project will have an impact on human resource development.