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.
The subject of this research project was the theory of functions,which is a part of pure mathematics. This means that the choice of problems of investigation was motivated by the inner logic of development of mathematics, rather than immediate applications. However, experience shows that the results of such fundamental research are indispensable for the progress of science and technology. Meromorphic functions is a class of mathematical functions which is most important in science and engineering. It includes most of the elementary functions as well as most of the special functions of mathematical physics. 1. Principal mathematical results of this project resulted in 25 publications in specialized mathematical journals. The most important results are the following: a) Complete proof of the conjecture of George Polya on the roots of successive derivatives of real entire functions. This conjecture was stated in 1943.This proof completes a half century of the efforts of several expert mathematicians. b) Finding the exact rate of approximation of certain discontinuous functions by polynomials of given degree. Approximation of functions by polynomials is important in several areas of technology, in particular, in electric engineering. This particular result already found applications in computer science. c) Establishing certain qualitative properties of energy levels of anharmonic oscillators. These problems were motivated by foundations of quantum mechanics,which is the most fundamental theory about the real world that modern science possesses. Energy levels of anharmonic oscillators have been intensively studied for about a century since the introduction of quantum mechanics, and a new method of their investigation has been developed within this project. 2. Impact on education. The Principal investigator advised 2 PhD students and one post-doctoral fellow. One of the students completed his PhD thesis in April 2012, and the second one is scheduled to complete in the end of the same year. Results of this research were used in two advanced courses that the PI taught to graduate students at Purdue University. 3. Outreach activities. The PI maintains web pages at www.math.purdue.edu/~eremenko which contain various resources on mathematics and science, oriented on a wide audience from school children, undergraduate students and amateurs of mathematics to researchers. One of these pages was a base of a radio broadcast in the series "Engines of our ingenuity", episode 2703. It has been also translated into Polish and Romanian languages, which gives some evidence of its popularity.