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