Diophantine approximation, Nevanlinna theory, and jet differentials

研究者从类比全纯曲线的值分布理论（Nevanlinna理论）的角度研究丢芬图问题，反之亦然。具体来说，一个实质性的组成部分将包括代数几何中关于射流空间理论的一般工作。然后，这项工作将应用于以下一个或多个主题：(i)在任意格式的一般背景下寻找Semple-Demailly商射流空间的适当推广，（ii）对数格式上的对数射流空间理论的公式，（iii） Yamanoi对“1+epsilon”猜想的证明的研究，以及（iv）寻找对数导数引理的数论对应物。该项目涉及数论（特别是整数或有理数多项式方程的解，以及相关不等式）和值分布理论（由一个或多个复变量的幂级数函数满足的不等式）。这两个看似不同的数学领域有许多共同的现象。这在一定程度上可以通过两个领域之间的形式类比来解释，这种类比导致了新的猜想、现有定理的简短证明和新的定理。本项目使用这个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.