Diophantine approximation, Nevanlinna theory, and jet differentials

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The investigator studies diophantine problems from the point of view ofthe analogy with the value distribution theory of holomorphic curves(Nevanlinna theory), and vice versa. Specifically, a substantialcomponent will consist of general work in algebraic geometry on thetheory of jet spaces. This work will then be applied to one or moreof the following topics: (i) work on finding a suitable generalization of Semple-Demailly quotient jet spaces to the general context of an arbitrary morphism of schemes, (ii) formulation of a theory of log jet spaces over log schemes, (iii) investigation of Yamanoi's proof of the "1+epsilon" conjecture, and (iv) a search for a number-theoretic counterpart to the lemma on the logarithmic derivative.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.