Pluri-Potential Theory and Geometric Function Theory

本项目提出了对复杂分析中各种问题的进一步研究。该提议包括两个主要部分：多势理论和几何函数理论。对于多变量的复杂分析，多势理论比一维分析的经典势理论更重要，因为随着维度的增加，使用分析工具变得更加困难。这就是为什么这个理论在复数分析的所有领域都有应用，包括复动力学，而且远远超出了它在代数和超越数论中的应用。几何函数理论的目标是在定量方法失败时获得被研究对象的定性特征，该理论在该项目中以区域、圆盘包络和自同构群的多势紧化的建议研究为代表。该项目还打算研究整函数环的代数性质及其在超越数论中的应用。在这个项目中，将进行几何复分析和数论的研究。几何作为数学的一部分，旨在描述不同对象之间的定性联系。例如，平行线不相交，三角形中的高度在同一点相交。当欧几里得对象：点、线和面被函数、曲面和集合等更复杂的结构取代时，研究就有了更微妙的味道。之所以会发生这种情况，是因为提供对象之间链接的机制不透明。在我们的提案中，我们将以类似于电荷能级的势的形式来寻找这样一种机制。我们还打算更好地理解数p的性质。

This project proposes the further study of various problems in complex analysis. There are two main parts to the proposal: the pluripotential theory and the geometric function theory. The pluripotential theory is more important for complex analysis in several variables than the classical potential theory for the one-dimensional analysis because it becomes more difficult to use analytic tools as the number of dimensions grows. That is why this theory found applications in all areas of complex analysis, including complex dynamics, and far beyond it in algebraic and transcendental number theories. The geometric function theory, whose goal is to obtain qualitative characteristics of investigated objects when quantitative approach fails, is represented in the project bysuggested studies of the pluripotential compactifications of domains, disk envelopes and groups of automorphisms. The project also intends to study algebraic properties of the ring of entire functions with applications to transcendental number theory.In this project research will be conducted on geometric complex analysis and number theory. Geometry, as a part of mathematics, aims to describe qualitative links between different objects. For example, parallel lines do not meet and the heights in a triangle meet at the same point. When Euclidean objects: points, lines and planes are replaced by more complicated structures like functions, surfaces and sets, the research is of a more delicate flavor. It happens because the mechanism providing links between objects is not transparent. In our proposal we will look for such a mechanism in the form of potentials similar to the energy levels of electrical charges. We also intend to understand better the properties of the number p.