Diophantine Approximation and Nevanlinna Theory

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

The investigator studies diophantine approximation from the point of view of the analogy with value distribution of holomorphic curves (Nevanlinna theory), and vice versa. Specifically, this work includes: (i) work on relating Faltings' 1983 theorem on the Shafarevich conjecture to Thue's method, (ii) work on finding further applications of the geometric logarithmic derivative lemma, (iii) a search for a number-theoretic counterpart to the lemma on the logarithmic derivative, and (iv) technical improvements in proofs using Thue's method (Dyson's lemma). This project involves number theory (specifically, solutions of polynomial equations in integers or rational numbers, as well as related inequalities) and value distribution theory (inequalities satisfied by power series functions of one or several complex variables). These two seemingly disparate areas of mathematics have many phenomena in common. This has been partly explained by a formal analogy between the two fields, which has led to new conjectures, shorter proofs of existing theorems, and new theorems. The present project uses this analogy to further elucidate the underlying structure of number theory, value distribution theory, and the analogy between the two fields.