Diophantine Approximation and Nevanlinna Theory

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The investigator studies diophantine approximation from the point ofview of the analogy with value distribution of holomorphic curves (Nevanlinnatheory), 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 polynomialequations in integers or rational numbers, as well as related inequalities)and value distribution theory (inequalities satisfied by power seriesfunctions of one or several complex variables). These two seeminglydisparate areas of mathematics have many phenomena in common. This hasbeen partly explained by a formal analogy between the two fields, whichhas led to new conjectures, shorter proofs of existing theorems, and newtheorems. The present project uses this analogy to further elucidatethe underlying structure of number theory, value distribution theory,and the analogy between the two fields.