Topics in extremal problems in complex analysis and potential theory

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The goal of the project is to provide new theoretical insight regarding deep extremal problems in complex analysis and potential theory. Particular emphasis will be placed on the method of symmetrization and the method of quadratic differentials, new versions of which were developed by the PI in his previous works. The PI will test these innovative techniques, in combination with a rich variety of existing tools, on some old challenging problems as well as on some new interesting questions posed in recent publications. The three main themes of the project are: (i) Developing covering theorems of Landau type and minimal area problems for analytic functions; (ii) Obtaining new sharp estimates for conformal invariants (such as capacities, harmonic measure, and hyperbolic density) of planar configurations with possible applications to problems on the boundary behavior of conformal mappings such as a well known Brennan's conjecture; (iii) Searching for a method to establish symmetry of zeros or critical points of extremal polynomials in some well known problems on complex polynomials such as Sendov's conjecture and Smale's conjecture. The PI anticipates that this study will involve the development of new approaches, which in turn will enhance our understanding of the theory of extremal problems in complex analysis and potential theory.The proposed methodology is targeted towards developing innovative tools in complex analysis which will have potential applications to other specific areas in science and engineering on at least two fronts. First, results in the theory of quadratic differentials impacts other branches of mathematics and has applications in theoretical physics, in particular, String Theory. Second, accurate estimates of functional characteristics of planar configurations, such as half-plane capacity, torsional rigidity, and principal frequency, are important in the theory of conformally invariant processes and in mathematical physics. The PI is working with an expanding core group of engaged and maturing graduate students. He expects the project, especially the problems about complex polynomials, which although elementary to state are very deep, to provide interesting research topics for them. Because Texas Tech University is geographically situated in rural West Texas with its under-served and under-represented populations, one of the opportunities for this project with its engagement of graduate students will be to attract more diverse and better qualified graduate students to careers in the mathematical sciences from the surrounding populations and ease placing them in good positions upon graduation.