NSF Award
0100512
The proposer intends to continue his study of geometric questions <br/>in the theory of meromorphic functions, using the new techniques <br/>developed in his previous work. The main directions of the <br/>proposed research are the following.<br/>a) Questions related to Bloch's theorem and the Type Problem of<br/>a simply connected Riemann surface, especially the relations<br/>between the conformal type of a surface and its integral curvature.<br/>b) Problems of geometric function theory arising in real algebraic<br/>geometry. More specifically, it includes counting real solutions<br/>of certain systems of algebraic equations of geometric origin, which<br/>have important applications in linear control theory. <br/>c) Generalization of results of geometric function theory to<br/>quasiregular maps in spaces of arbitrary dimension.<br/>d) Normality criteria for families of holomorphic curves in projective<br/>spaces.<br/><br/>One of the basic questions in mathematics and its applications is<br/>whether a given equation or a system of equations has solutions,<br/>how many, and where are they located. In the theory of meromorphic functions<br/>one studies these questions for equations of the type<br/>f(z)=a, where a is a given complex number and f a given meromorphic function.<br/>The class of meromorphic functions includes elementary functions,<br/>such as rational, exponential and trigonometric ones, as well as the special<br/>functions, a. k. a. higher transcendental functions, such as the<br/>Gamma function, Airy functions, elliptic functions and so on.<br/>Most functions arising in applications of mathematics belong to<br/>this class. In modern mathematics, questions about solvability<br/>of equations are usually formulated in geometric language, which makes<br/>the results appealing to our geometric intuition. <br/>The logic of development of mathematics and its applications<br/>require an extension of results to vector-valued functions known<br/>as ``holomorphic curves''. The proposer plans to continue his study<br/>of geometric theory of meromorphic functions and holomorphic curves.<br/>A part of the proposal is related to existence of real solutions, which is<br/>by far more subtle than the existence of complex solutions, which are usually<br/>studied. This part is inspired by the so-called "pole placement problem", which is a major unsolved mathematical<br/>problem in control theory of linear systems. The results in this area<br/>will have implications for the design of complicated automatic control systems. <br/>These results would establish limitations on the possibility to<br/>control a system of given size by a control device of certain class.
