Analysis on Berkovich spaces and applications

这个建议是关于Berkovich空间理论的进一步发展，以及算术应用。从广义上讲，PI计划做以下工作：(1)发展度量图的雅可比矩阵理论，并应用于研究Berkovich解析曲线与其雅可比矩阵之间的关系；(2)将Berkovich投影线上的PI势理论和Rumely势理论推广到高维；(3)利用Berkovich空间理论，对算法动力学中的一些开放问题取得了新的进展。所提出的研究方法涉及到算术几何、分析、拓扑学、图论和动力系统等技术的混合。从广义上讲，该提案旨在解决数论中的一个关键统一原则，即全局域的所有补全都应该以对称的方式处理。近一百年来，这一直是数论的中心主题，例如，切瓦莱（Chevalley）对类场论（class field theory）的理想化表述和泰特（Tate）对阿德尔斯（adeles）的谐波分析的发展就说明了这一点。这项工作的主要智力价值是它将导致伯科维奇空间理论的新发展，这是一个令人兴奋的主题，已经在数论（包括局部朗兰兹对应），数学物理（例如镜像对称）和其他不同领域找到了应用。该项目还将展示如何将Berkovich空间应用于与动力系统相关的各种有趣问题。此外，PI将给度量图的理论带来新的思路。度量图，在文献中也被称为度量图或量子图，应用于各种领域，包括物理学、数学生物学和数论。拟议工作的更广泛影响将包括研究和说明性论文的传播，与不同领域的专家的互动，以及对本科生、研究生和博士后研究的支持。

DMS-0600027Matthew BakerThis proposal is concerned with further development of the theory of Berkovich spaces, together with arithmetic applications. In broad terms, the PI plans to do the following: (1) Develop a theory of Jacobians of metrized graphs, with applications to studying the relationship between a Berkovich analytic curve and its Jacobian; (2) generalize the PI's and Rumely's potential theory on the Berkovich projective line to higher dimensions; and (3) use the theory of Berkovich spaces to make new progress on some open problems in arithmetic dynamics. The methods to be employed in the proposed research involve a mixture of techniques from arithmetic geometry, analysis, topology, graph theory, and dynamical systems. In broad terms, the proposal aims to address a key unifying principle within number theory, the idea that all completions of a global field should be treated in a symmetric way. This has been a central theme in number theory for almost a hundred years, as illustrated for example by Chevalley's idelic formulation of class field theory and Tate's development of harmonic analysis on adeles.The main intellectual merit of this work is that it will lead to new developments in the theory of Berkovich spaces, an exciting subject which has already found applications to number theory (including the local Langlands correspondence), mathematical physics (e.g. mirror symmetry), and other diverse fields. This project will also show how Berkovich spaces can be applied to a variety of interesting questions related to dynamical systems. In addition, the PI will bring new ideas to the theory of metrized graphs. Metrized graphs, also known in the literature as metric or quantum graphs, have applications to a diverse array of fields, including physics, mathematical biology, and number theory. The broader impacts of the proposed work will include dissemination of research and expository papers, interaction with experts in different fields, and support for undergraduate, graduate, and postdoctoral research.