Mathematical Sciences: Diophantine Approximations of Algebraic Points

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This project deals with diophantine approximation and value distribution theory of holomorphic functions. Work by P. Vojta, C. Osgood, and others has shown that these two fields are in fact connected by a formal analogy. This observation has led to new conjectures, shorter proofs of old theorems, and proofs of new theorems. This project will continue this trend, primarily working in the field of diophantine approximation. Results whose proofs may be simplified include finiteness theorems for rational points on abelian varieties and Faltings' theorem on the finiteness of principally polarized abelian varieties over a given field with finite reduction. New results which may be proved include "moving targets" type results; i.e., diophantine approximation statements which involve a slowly varying family of equations. A distant goal is the "abc conjecture." Technical improvements in proofs using Thue's method will also be pursued. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.