Analysis on Berkovich spaces and applications

DMS-0600027马修·贝克这一建议涉及到进一步发展的理论，伯科维奇空间，以及算术应用。 广义而言，PI计划做以下工作：（1）发展度量化图的Jacobian理论，并应用于研究Berkovich解析曲线与其Jacobian之间的关系;（2）将PI和Rumely关于Berkovich投影线的势理论推广到更高维;（3）利用Berkovich空间理论对算术动力学中的一些开放问题进行了新的探索。 拟采用的方法在拟议的研究涉及算术几何，分析，拓扑，图论和动力系统的技术的混合物。 从广义上讲，这个提议旨在解决数论中一个关键的统一原则，即一个整体域的所有完备化都应该以对称的方式处理。 这一直是一个中心主题，在数论近百年来，作为说明，例如Chevalley的idelic制定类域理论和泰特的发展调和分析adeles.The主要智力价值这项工作是，它将导致新的发展理论的Berkovich空间，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）和其他不同领域中找到了应用。 这个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的问题。此外，PI将给度量化图理论带来新的思路。度量化图（Metrized graphs），在文献中也被称为度规图或量子图，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临时论文的传播，与不同领域专家的互动，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.