Berkovich Spaces, Tropical Geometry, and Arithmetic Dynamics

这项建议涉及一系列不同主题的问题，包括伯克维奇空间、热带几何学和复杂动力学。这项建议的主要学术价值在于，它将增加我们对这些重要数学领域的理解，并揭示它们之间的新关系。这些问题背后的主要统一主题是，我们提出的解决这些问题的策略都涉及势能理论，无论是在经典的还是非阿基米德的背景下。近年来，PI和其他人发展了一种令人惊讶的强健的非阿基米德经典复势理论模拟。此外，PI还帮助发展了一些用于比较代数簇的Berkovich分析和热带化的一般技术，表明人们可以有利可图地将热带几何视为Berkovich的非阿基米德解析空间理论和经典凸几何之间的‘桥梁’。PI提出了通过热带几何构造曲线的半稳定模型的新方法，证明了Mumford-Neeman等分布定理的非阿基米德Berkovich空间版本，将Berkovich理论应用于Nelon模型的分量群的研究，并在所有有理映射的模空间内探索后临界有限有理映射的算术和几何性质。复势理论的经典主题最早出现在物理学中，用于描述引力和电磁相互作用。随后，它在数学研究的各个领域得到了大量的应用，包括复杂分析和复杂动力学(它被用来研究分形学，如著名的曼德布罗特集)。非阿基米德分析是现代数论的重要组成部分，最早出现在二十世纪初亨赛尔关于著名的p-进数论的著作中。在非阿基米德位势理论中，人们用一个称为Berkovich投影线的p元对应的复数来代替经典的Riemann球，这是由Vladimir Berkovich在1980年出版的《S》一书中提出的。热带几何是一个相对较新和活跃的研究领域，应用于许多数学领域。人们可以把热带几何看作是对经典代数几何的分段线性近似，在该几何中，用多面体复数(被认为是线性不等式系统的公共解的集合)来代替“代数族”(粗略地说，这是一组多项式方程的公共解)。令人惊讶的是--也是相当神秘的--热带近似值比人们最初预期的要记住更多关于原始物种的信息。

This proposal involves problems in a diverse array of topics including Berkovich spaces, tropical geometry, and complex dynamics. The primary intellectual merit of the proposal is that it will increase our understanding of each of these important areas of mathematics and unearth new relationships between them. The main unifying theme behind these problems is that our proposed strategies for solving them all involve potential theory, both in the classical and non-Archimedean setting. In recent years, a surprisingly robust non-Archimedean analog of classical complex potential theory has been developed by the PI and others. In addition, the PI has helped to develop a number of general techniques for comparing Berkovich analytifications and tropicalizations of algebraic varieties, showing that one can profitably view tropical geometry a `bridge' between Berkovich's theory of non-Archimedean analytic spaces and classical convex geometry. The PI proposes to develop new methods for constructing semistable models of curves via tropical geometry, to prove a non-Archimedean Berkovich space version of the Mumford-Neeman equidistribution theorem, to apply Berkovich's theory to the study of component groups of Neron models, and to explore arithmetic and geometric properties of post-critically finite rational maps within the moduli space of all rational maps.The classical subject of complex potential theory first arose in physics, where it was used to describe gravitational and electromagnetic interactions. It has subsequently found a wealth of applications to various areas of mathematical research, including complex analysis and complex dynamics (where it is used to study fractals such as the celebrated Mandelbrot set). Non-Archimedean analysis is a crucial part of modern number theory which first arose in the early twentieth century work of Kurt Hensel on the famous 'p-adic numbers'. In non-Archimedean potential theory, one replaces the classical complex ``Riemann sphere'' by a p-adic counterpart, called the Berkovich projective line, which was introduced by Vladimir Berkovich in the 1980's. Berkovich's theory has since become an important tool in modern number theory and algebraic geometry. Tropical geometry is a relatively new and active area of research with applications to many fields of mathematics. One can think of tropical geometry as a piecewise linear approximation of classical algebraic geometry in which an ``algebraic variety'' (which is, roughly speaking, the set of common solutions to a system of polynomial equations) is replaced by a polyhedral complex (thought of as the set of common solutions to a system of linear inequalities). Surprisingly -- and rather mysteriously -- the tropical approximation remembers much more information about the original variety than one might originally expect.