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.
One of major achievements during my work supported by this NSF grant was completion of my research project on Continuous Symmetrization. Specifically, I introduced a new geometric transformation which changes shape of a planar or solid body continuously in such a way that all basic characteristics are changing monotonically. For instance, this transformation allows continuously transform bodies into symmetric configurations (in particular into balls) with the same volume in such a way that their capacity and corresponding energy (required to support this configuration) will decrease all the time during this process while the torsional rigidity of the body , which characterizes how strong is this configuration, will increase. My publication on this subject (which is 50 pages long) received very positive response from analysts. In particular, the reviewer of this work for the Mathematical Reviews of the American Mathematical Society wrote: "The present paper is well worth reading/discussing/studying, both for the content and for the presentation of the abundance of material" and "The material presented in the paper under review is substantial enough for a monograph". Another important outcome of the project is advances in the so-called "Iceberg-type problems". In these problems the whole configuration consists of two parts, visible and hidden. Then the main question is to estimate characteristics of the hidden part from the knowledge of characteristics of the visible part. We did not consider real iceberg. Mathematically, these problems are boundary values problems with partially free/fixed boundary. In a joint paper with my collaborators, we complete solved the problem on the maximal draft of the mathematical iceberg and made some progress in the problems on the maximal width of the iceberg and on the minimal safe distance from the iceberg. Among other outcomes from this project, I want to mention my solutions of two extremal problems posted by recognized leaders in my area of research. I believe that publishing solutions of such difficult problems in prestigious journals will increase visibility of our Department of Mathematics and Statistics and visibility of Texas Tech University, in general. This may help to attract more good students to our graduate program. One of my students made a presentation at the session Family Track Activities, which is a part of the 2013 MOVES Conference, in New York City, the goal of which is to attract talented kids and their families to mathematics. Although I did not have support for my students from this NSF grant, during this period three of my PhD students completed their projects on the problems related to my research sponsored by this NSF grant and obtained their PhD degrees from Texas Tech. Currently I am directing or co-directing research projects of five PhD students.
