Real meromorphic functions

Purdue University, West Lafayette IN

Investigators

Abstract

ABSTRACT The proposer intends to continue his study of the distribution of roots and critical points of real meromorphic functions using the geometric methods developed in his previous work. The main directions of the proposed research are the following. a) The study of the Wronski map, both in the real and complex domains, and of the related pole assignment map. b) The study of the distribution of roots of successive derivatives of real entire functions. c) Further investigation of the relation between the rate of oscillation of real functions and their spectral properties. d) The study of existence and uniqueness of metrics of positive curvature with conic singularities on compact surfaces. 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 resolvability of equations are usually formulated in geometric language, which makes the results appealing to our geometric intuition. The proposer plans to continue his study of geometric theory of meromorphic functions. Most of the proposed research is related to existence of real solutions, a moresubtle question than the existence of complex solutions, which are usually studied. One of the original motivations (beside intrinsic mathematical importance of these questions) was 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. Another important area of the broader impact is the recently discovered connection of the problems considered in this proposal with physics, more precisely, with the exactly solvable models of ferromagnetism.