Problems in geometric function theory

Purdue University, West Lafayette IN

Investigators

Abstract

Analytic functions are objects with very nice properties that allow one to use a multitude of techniques to explore them, and they constitute the most important class of functions used in mathematics and its applications to science and engineering. The choice of problems of the theory of analytic functions in this proposal is motivated by their intrinsic mathematical interest, rather than specific applications. The problems chosen for this proposal are old and difficult, and the PI proposes to study them using a variety of new methods recently developed by him and his collaborators, as well as by the other research groups. Most of these problems arise in the previous research of the PI funded by NSF. This previous research led to several significant results relevant to applications of mathematics in control theory, physics and material science. They also had impact on education: advanced mathematical courses on graduate level, PhD theses supervised by the PI and training of postdoctoral scholars. The present proposal is expected to have a similar impact. The main topic of this proposal is the geometric problem of classification of spherical polygons, and more generally, of metrics of constant positive curvature with conical singularities. This is closely related to the study of analytic functions defined by the Heun equation, the linear second order differential equation with four regular singularities. These functions lie on the boundary of the set of Special functions of Mathematical physics, but much less is known about them in comparison with classical special functions which are defined by equations with at most three singularities. The goal of this research is to classify spherical quadrilaterals up to isometry. The methods proposed are those used in the recent research of PI, A. Gabrielov, E. Mukhin, V. Tarasov and A. Varchenko on the Shapiro conjecture in real algebraic geometry.The second subject of proposed research is value distribution of holomorphic curves in projective spaces, especially of Brody curves, and curves defined by higher order linear differential equations. One specific goal is to prove a recent conjecture of Duval and da Costa on the Second Main theorem for Brody curves. The PI plans to use potential-theoretic methods developed in his previous work. The third topic is investigation of a class of extremal problems of new type for holomorphic functions in the unit disc that are inspired by questions in Control theory, namely the question of stabilizability of several systems by single output feedback control device. The proposed methods are those employed in the previous research on this problem by PI, jointly with W. Bergweiler.