Berkovich Spaces, Tropical Geometry, and Arithmetic Dynamics

这项建议涉及的问题，在不同的主题，包括布氏空间，热带几何，复杂的动力学。 这个建议的主要智力价值是，它将增加我们对数学的每个重要领域的理解，并挖掘它们之间的新关系。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潜在的理论，无论是在经典和非阿基米德设置。 近年来，PI和其他人开发了一种令人惊讶的经典复势理论的非阿基米德模拟。 此外，PI已帮助开发了一些一般技术比较Berkovich analytifications和tropicalizations的代数品种，显示出一个可以获利查看热带几何之间的“桥梁”Berkovich的理论非阿基米德解析空间和经典凸几何。 PI建议开发新的方法来通过热带几何构造曲线的半稳定模型，证明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，并探索后-在所有有理映射的模空间中的临界有限有理映射。经典的复势理论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中得到了大量的应用，包括复分析和复动力学（它被用来研究分形，如著名的曼德尔布罗特集）。 非阿基米德分析是现代数论的一个重要组成部分，它首先出现在世纪早期库尔特·亨泽尔关于著名的“p进数”的工作中。 在非阿基米德势理论中，人们用一个称为伯科维奇投影线的p-adic对应物取代了经典的复数"黎曼球“，该投影线是由弗拉基米尔·伯科维奇在20世纪80年代提出的。 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和活跃的研究领域，应用于许多数学领域。人们可以认为热带几何是经典代数几何的分段线性近似，其中“代数变量”（粗略地说，是多项式方程组的公共解的集合）被多面体复形（被认为是线性不等式系统的公共解的集合）取代。 令人惊讶的是--而且相当神秘的是--热带近似记忆了比人们最初预期的要多得多的关于原始品种的信息。

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.