Polynomial Inequalities and Applications

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This research project concerns mathematical analysis, in particular in approximation theory, orthogonal polynomials, and potential theory. These are parts of mathematics that originated well over a century ago and have provided the foundations and the tools for widespread applications, from magnetic resonance imaging scanners to airplane designs. One main focus of the work is to understand complicated mathematical objects in terms of relatively transparent and computable quantities. The project is a continuation of this classical area, but with a new and fresh look at some of its questions, aimed at developing new methods to solve some well-known open problems. Though it is basic research, the results of the project are anticipated to be useful in other areas of mathematics, physics, and engineering. It is expected that the project will stimulate interest in undergraduates and enhance the research environment for them. There are four main research areas that are considered. The first is the approximative extensions of conformal maps/Green's functions when conventional conformal extensions do not exist. These can be used in lieu of conformal extensions when less smoothness than analyticity is assumed. The second area concerns new types of polynomial inequalities to settle the best constant problem in Bernstein- and Markov-type inequalities on sets consisting of smooth Jordan curves and arcs, thereby closing a chapter in approximation theory that is more than a century old. The third is the study of Christoffel-Darboux kernels associated with general families of orthogonal polynomials with a possible application to universality results in random matrix theory. The fourth research area concerns the clarification and resolution of Widom's conjecture on the norm of Chebyshev polynomials. Various polynomial inequalities connect these fields and are expected to play a decisive role in the solutions of the problems under study.