Diophantine approximation, Nevanlinna theory, and jet differentials

研究者从与全纯曲线的值分布理论（Nevanlinna理论）相似的角度研究丢番图问题，反之亦然。 具体来说，一个substantialcomponent将包括一般工作代数几何理论的喷流空间。 本文的工作将应用于以下一个或多个主题：（i）寻找Semple-Demailly商的适当推广 喷射空间的一般情况下，任意态射的计划，（ii）制定的理论，日志喷射空间的日志计划，（iii）调查山野井的证明“1+ 1”猜想，和（iv）搜索数论对应的引理上的 该项目涉及数论（具体而言，整数或有理数中多项式方程的解以及相关的不等式）和值分布理论（一个或多个复变量的幂级数函数满足的不等式）。 这两个截然不同的数学领域有许多共同的现象。 这已经部分地解释了两个领域之间的形式类比，这导致了新的假设，现有定理的较短证明和新定理。 本项目使用这种类比来进一步阐明数论、值分布理论的基本结构以及这两个领域之间的类比。

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.