Berkovich Spaces, Tropical Geometry, and Arithmetic Dynamics

该方案涉及各种主题的问题，包括伯科维奇空间、热带几何和复杂动力学。这一建议的主要智力价值在于，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到势理论，无论是在经典的还是在非阿基米德的背景下。近年来，PI和其他人开发了一个令人惊讶的经典复势理论的非阿基米德模拟。此外，PI还帮助开发了一些通用技术，用于比较Berkovich分析和代数变量的热带化，表明人们可以将热带几何视为Berkovich非阿基米德解析空间理论与经典凸几何之间的“桥梁”。PI提出了通过热带几何构造曲线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Neron模型的组成群的研究，并在所有有理映射的模空间内探索后批判有限有理映射的算术和几何性质。复势理论的经典主题首先出现在物理学中，它被用来描述引力和电磁相互作用。随后，它在数学研究的各个领域找到了丰富的应用，包括复杂分析和复杂动力学（用于研究分形，如著名的Mandelbrot集合）。非阿基米德分析是现代数论的重要组成部分，它最早出现在20世纪早期库尔特·亨塞尔关于著名的p进数的研究中。在非阿基米德势理论中，人们用p进的对应物取代经典复合体“Riemann sphere”，称为Berkovich投影线，它是由Vladimir Berkovich在1980年代引入的。伯科维奇的理论从此成为现代数论和代数几何的重要工具。热带几何是一个相对较新的活跃的研究领域，在数学的许多领域都有应用。人们可以把热带几何看作是经典代数几何的分段线性逼近，其中的“代数变化”（粗略地说，是多项式方程系统的公解集）被多面体复合体（被认为是线性不等式系统的公解集）所取代。令人惊讶的是，而且相当神秘的是，热带近似记住的原始品种的信息比人们最初预期的要多得多。

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.