Analysis on Berkovich spaces and applications

Berkovich空间分析及应用

• 批准号: 0600027
• 负责人: Matthew Baker
• 金额: --
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2010-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600027&HistoricalAwards=false
• 关键词: Analysis; Berkovich; spaces; applications

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.

DMS-0600027Matthew Baker该提案涉及伯科维奇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) 开发度量图的雅可比行列式理论，并应用于研究伯科维奇解析曲线与其雅可比行列式之间的关系； (2)将Berkovich射影线上的PI和Rumely势理论推广到更高维度； (3)利用伯科维奇空间理论在算术动力学的一些开放性问题上取得新进展。 拟议研究中采用的方法涉及算术几何、分析、拓扑、图论和动力系统技术的混合。 从广义上讲，该提案旨在解决数论中的一个关键统一原则，即全球领域的所有完成都应以对称方式对待。 近一百年来，这一直是数论的中心主题，例如谢瓦利（Chevalley）的类场论的理想表述和泰特（Tate）对阿黛尔（adeles）调和分析的发展就说明了这一点。这项工作的主要智力价值在于它将带来新的知识。伯科维奇空间理论的发展是一个令人兴奋的学科，它已经应用于数论（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。 该项目还将展示如何将伯科维奇空间应用于与动力系统相关的各种有趣问题。此外，PI还将给度量化图理论带来新的思路。度量图（在文献中也称为度量图或量子图）可应用于多种领域，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传播研究和说明性论文、与不同领域的专家互动以及对本科生、研究生和博士后研究的支持。