Geometric Theory of Meromorphic Functions

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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 ofa simply connected Riemann surface, especially the relationsbetween the conformal type of a surface and its integral curvature.b) Problems of geometric function theory arising in real algebraicgeometry. More specifically, it includes counting real solutionsof certain systems of algebraic equations of geometric origin, whichhave important applications in linear control theory. c) Generalization of results of geometric function theory toquasiregular maps in spaces of arbitrary dimension.d) Normality criteria for families of holomorphic curves in projectivespaces.One of the basic questions in mathematics and its applications iswhether a given equation or a system of equations has solutions,how many, and where are they located. In the theory of meromorphic functionsone studies these questions for equations of the typef(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 specialfunctions, a. k. a. higher transcendental functions, such as theGamma function, Airy functions, elliptic functions and so on.Most functions arising in applications of mathematics belong tothis class. In modern mathematics, questions about solvabilityof equations are usually formulated in geometric language, which makesthe results appealing to our geometric intuition. The logic of development of mathematics and its applicationsrequire an extension of results to vector-valued functions knownas ``holomorphic curves''. The proposer plans to continue his studyof geometric theory of meromorphic functions and holomorphic curves.A part of the proposal is related to existence of real solutions, which isby far more subtle than the existence of complex solutions, which are usuallystudied. This part is inspired by the so-called "pole placement problem", which is a major unsolved mathematicalproblem in control theory of linear systems. The results in this areawill have implications for the design of complicated automatic control systems. These results would establish limitations on the possibility tocontrol a system of given size by a control device of certain class.