Entire and Meromorphic Functions in Several Complex Variables

该研究针对全全纯映射上的复变量问题、内插问题和值分布理论，提出继续研究关于整个分析集体积增长的先验Bezout问题，并对全全纯映射的零点的计数函数进行精确的估计。他还建议研究整函数的加权空间的内插问题，这是多复变量中的基本和中心问题之一。特别是，他希望对插值变元进行几何刻画，这与他对Bezout问题的研究密切相关。在第三个方向上，他想继续他对值分布理论的研究，特别是关于慢运动目标的精化的内夫林纳第二基本定理及其与其他问题的关系，如亚纯函数的唯一性问题和偏微分方程亚纯解的问题。上述问题不仅从几个复变量的角度来看是重要的，而且从它们的关系和应用到其他学科，如调和分析，超越数论，系统论和工程学，都是重要的。例如，许多重要的问题，如寻找和表示信号处理、图像压缩、材料统计等中出现的偏微分方程组或卷积方程组的解，等价于加权空间中整函数的插补问题。本研究旨在发展和推进上述几个复杂变量领域的理论和应用。

The proposed research is directed at topics in several complex variables onentire holomorphic maps, interpolation problems, and value distribution theory.The principle investigator proposes to continue his study on the well-knowntranscendental Bezout problem about the growth of the volume of entire analyticsets and find sharp estimates on the counting functions of zeros of entireholomorphic maps. He also proposes to study interpolation problems for weighted spaces of entire functions, one of fundamental and central problems in several complex variables. In particular, he wishes to characterize interpolatingvarieties geometrically, which is closely tied with his study on the Bezoutproblem. In the third direction, he would like to continue his research on valuedistribution theory with a special attention given to the refined Nevanlinnasecond fundamental theorem for slowly moving targets and its relations to otherproblems such as uniqueness problems of meromorphic functions and meromorphic solutions of partial differential equations.The above problems are important not just from the point of view of severalcomplex variables, but also from their relations and applications to othersubjects such as harmonic analysis, transcendental number theory, systems theory and engineering. For instance, many important problems like finding andrepresenting solutions to partial differential equations or systems ofconvolution equations arising in signal processing, image compression, materialstesting, etc. are equivalent to interpolation problems for entire functions inweighted spaces. The proposed research aims at developing and advancing boththeories and applications for the above areas in several complex variables.