Meromorphic Functions and Holomorphic Curves

Proposal: DMS-9800084 Principal Investigator: Alexandre Eremenko Abstract: A. Eremenko will continue his study of the distribution of values of meromorphic functions and their generalization, holomorphic maps from the complex line to projective spaces, using the new methods of potential theory developed in his previous work. With this approach he already obtained such results as the proof of a defect relation for non-linear hypersurfaces conjectured by B. Shiffman (joint work with Sodin, 1992) and the proof of the Modified Cartan conjecture in dimension three (1996). Specifically, he plans to concentrate on the following topics: normality criteria for holomorphic maps omitting hyperplanes, the study of extremal maps for defect relations, and the Inverse Problem of the Value Distribution Theory for maps to projective spaces, with special attention to the maps satisfying algebraic differential equations, both linear and non-linear. Meromorphic functions constitute the most basic class of functions used in mathematics and its applications. This class includes elementary functions like the exponential, cosine and tangent functions, as well as higher transcendental functions, like the Gamma function, Airy functions, elliptic functions, etc., which are indispensable in physics, engineering and other applications. The fundamental problem about these functions is the study of solutions of equations f(x)=a, where f is a given function, and a is a given complex number. Usually such equations have infinitely many solutions and the question is how their location depends on the right-hand side of the equation. This is the subject of Value Distribution Theory, a great achievement of mathematical analysis in the second half of the nineteenth century and the first half of the present century. The logic of development of mathematics and potential applications to differential equations require an extension of the main results of this theory to the case of functions whose values are no longer numbers, but i nstead are vectors. Despite major efforts by several outstanding researchers in the beginning of the twentieth century, our current understanding of this subject is very incomplete. Recent breakthroughs give hope of major new advances in the area. Such advances would extend our knowledge of a basic and fundamental mathematical subject, which is the value distribution of the vector-valued entire functions known as "holomorphic curves in projective spaces."

Proposal: DMS-9800084 Principal Investigator: Alexandre Eremenko Abstract: A. Eremenko will continue his study of the distribution of values of meromorphic functions and their generalization, holomorphic maps from the complex line to projective spaces, using the new methods of potential theory developed in his previous work. With this approach he already obtained such results as the proof of a defect relation for non-linear hypersurfaces conjectured by B. Shiffman (joint work with Sodin, 1992) and the proof of the Modified Cartan conjecture in dimension three (1996). Specifically, he plans to concentrate on the following topics: normality criteria for holomorphic maps omitting hyperplanes, the study of extremal maps for defect relations, and the Inverse Problem of the Value Distribution Theory for maps to projective spaces, with special attention to the maps satisfying algebraic differential equations, both linear and non-linear. Meromorphic functions constitute the most basic class of functions used in mathematics and its applications. This class includes elementary functions like the exponential, cosine and tangent functions, as well as higher transcendental functions, like the Gamma function, Airy functions, elliptic functions, etc., which are indispensable in physics, engineering and other applications. The fundamental problem about these functions is the study of solutions of equations f(x)=a, where f is a given function, and a is a given complex number. Usually such equations have infinitely many solutions and the question is how their location depends on the right-hand side of the equation. This is the subject of Value Distribution Theory, a great achievement of mathematical analysis in the second half of the nineteenth century and the first half of the present century. The logic of development of mathematics and potential applications to differential equations require an extension of the main results of this theory to the case of functions whose values are no longer numbers, but i nstead are vectors. Despite major efforts by several outstanding researchers in the beginning of the twentieth century, our current understanding of this subject is very incomplete. Recent breakthroughs give hope of major new advances in the area. Such advances would extend our knowledge of a basic and fundamental mathematical subject, which is the value distribution of the vector-valued entire functions known as "holomorphic curves in projective spaces."