Berkovich Spaces, Tropical Geometry, and Arithmetic Dynamics

伯科维奇空间、热带几何和算术动力学

• 批准号: 1201473
• 负责人: Matthew Baker
• 金额: $ 36万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201473&HistoricalAwards=false
• 关键词: Berkovich; Spaces; Tropical; Geometry; Arithmetic

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.

该提案涉及各种各样的主题中的问题，包括伯科维奇空间，热带几何形状和复杂的动态。 该提案的主要知识优势在于，它将增加我们对数学和发现它们之间的新关系中每个重要领域的理解。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在古典和非安置的环境中的潜在理论。 近年来，PI和其他人已经开发了一种令人惊讶的强大的经典复杂潜在理论的非一切集类似物。 此外，PI有助于开发许多代数品种的伯科维奇分析和热带化的一般技术，表明人们可以盈利地将热带几何形状视为伯科维奇的非架构分析空间和经典的互联网几何学之间的“桥梁”。 PI建议通过热带几何形状开发新方法，以证明Mumford-neeman等分定理的非架构的Berkovich空间版本，以将Berkovich的理论应用于Neron模型的组成组，以及对Neron模型的研究，以及探索在所有有理图的模量空间内批判性有限有限理性图的算术和几何特性。 随后，它在数学研究的各个领域中发现了大量应用，包括复杂的分析和复杂的动态（用于研究著名的Mandelbrot集）。 非一切现象分析是现代数字理论的关键部分，它在二十世纪初期的库尔特·汉塞尔（Kurt Hensel）作品首次出现，以著名的“ p-adic数字”。 在非架构的潜在理论中，一个人用P-ADIC对应物（称为Berkovich Projective Line）取代了古典复合体``Riemann Sphere''，该系列是由弗拉基米尔·伯科维奇（Vladimir Berkovich）在1980年代引入的。 此后，伯科维奇的理论已成为现代数字理论和代数几何形状的重要工具。 热带几何形状是一个相对较新的研究领域，并应用于许多数学领域。人们可以将热带几何形状视为经典代数几何形状的分段线性近似，其中``代数品种''（粗略地说，这是多面体复合物的一组通用解决方案）（被认为是线性不平等系统的一组通用解决方案）。 令人惊讶的是 - 而且很神秘 - 热带近似记得有关原始品种的信息，比人们最初预期的要多。