单与多复变量差分Nevanlinna理论及在复差分方程的应用

In this project we mainly study the Difference Nevanlinna theory in one and several complex variables and its applications on complex difference equations. This belongs to one of important subjects in modern complex analysis. The Nevanlinna was originated from 1920s, and has been investigated by many mathematician. Difference Nevnalinna theory has benn established since 2006 and causes an important development of differential difference equations again. It is widely focused on. Up to now, on one hand, difference Nevanlinna theory itself is developing fast, and has a lot of key problems need to be studied, furthermore, there is a direction on the development of the high dimensional case need to be digged; on the other hand, it needs to be systemly investigated of complex differential difference equations, especially the partial differential difference equations whic is very chanllenged. The applicant and main members have obtained revelatively well work, will combine with high dimensional Nevanlinna theory, several complex variables, geometry, partial differential equations and tropical mathematics, and now intend to do further research on some of the important issues: Nevanlinna theory of holomorphic maps concerning some differnce operatos; tropical version of difference Nevanlinna theory; system calssification of difference Panleve equations; Fermat type partial differential difference equations. We believe that we will make some breakthrough and obtain some interesting results.

本项目研究单与多复变量差分Nevanlinna理论及在复差分方程的应用，是复分析具有挑战前沿核心课题之一。Nevanlinna理论起源于19世纪20年代，国内外许多著名数学家都加入研究。自2006年始，差分Nevanlinna理论得以建立，带动了复微分差分方程理论迅速崛起，广受关注。目前，差分Nevnalinna理论中很多核心问题有待深入研究，高维情形急需挖掘；复微分差分方程理论虽发展迅速，但更需开拓，其中复偏微分差分方程理论鲜有人研究，富有挑战。申请人与主要成员已具备了较好的前期研究基础，拟结合高维Nevanlinna理论、多复变、几何、偏微分方程、tropical数学等，深入研究如下重要问题：若干差分算子的全纯映射Nevanlinna理论；Tropical差分Nevanlinna理论；差分Panleve方程的系统分类；Fermat型复偏微分差分方程。有望取得一些突破，获得重要意义成果。

本项目研究单与多复变量差分Nevanlinna理论及在复差分方程的应用，是复分析具有挑战前沿核心课题之一。Nevanlinna理论起源于19世纪20年代，国内外许多著名数学家都加入研究。自2006年始，差分Nevanlinna理论得以建立，带动了复微分差分方程理论迅速崛起，广受关注。差分Nevnalinna理论中很多核心问题有待深入研究，高维情形急需挖掘；复微分差分方程理论虽发展迅速，但更需开拓，其中复偏微分差分方程理论鲜有人研究，富有挑战。项目负责人与主要成员在具备了较好的前期研究基础，深入进行科研攻关，获得了重要学术成果，共计发表了25篇学术论文，其中19篇SCI收录。主要在以下几个方面取得了重要学术成果，部分达到了国际同行先进水平：.（1）对多复变亚纯函数的对数差分引理、q-差分算子对数导数引理进行了创新性地研究，并起始性应用于研究复偏差分微分方程；.(2)起始性地研究了Logistic型时滞微分方程亚纯解理论，将英国数学家Halburd和芬兰数学家Korhonen关于两个时滞推广到任意多个时滞的情形，推广了方程解零点和极点的迭代方法，并对高阶导数情形方程做了起始性研究；.(3)首次考虑了Jackson差分算子的值分布理论，获得了系列重要研究工作；.(4)对多复变亚纯函数的正规族理论、唯一性定理和差分算子的唯一性定理进行了重要研究；.(5) 深入考虑亚纯映射值分布理论，将Ji-Yan-Yu的超曲面次一般位置带有k指标的概念推广到移动超曲面目标情形和闭子概型情形，获得了到复代数簇及移动超曲面以及闭子概型的第二基本定理；.(6)对单复变差分方程理论进行了系列深入的研究。