The proposed research lies in the interface of several complex variables, number theory, and differential geometry, in particular, it involves the somewhat mysterious relationship between Nevanlinna theory and the theory of Diophantine geometry. The following topics will be investigated: moving target problems in Diophantine approximation and Nevanlinna theory, Diophantine approximation by algebraic points of bounded degree, and the Kobayashi conjecture in the theory of complex hyperbolic geometry. A second part of the project will involve the study of the value distribution properties of the Gauss map of minimal surfaces and minimal submanifolds in the n-dimensional Euclidean space.
Number theory and complex analysis, two old but important subjects, have been found to have important applications in Engineering, Biology, Computer Sciences, and other fields. Historically these two subjects developed independently. Recently it was discovered that they are somewhat related; in particular, Nevanlinna theory in complex analysis and Diophantine approximation in number theory bear striking similarities and connections. This interplay has lead to several important new results in number theory and in complex analysis. It is hoped that the newly discovered relationship will revolutionize the researches and will lead major new advances in both subjects. Such advances would have a great impact in the whole area of mathematics.
