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 Bakerthis提案与伯科维奇空间理论的进一步发展以及算术应用有关。 从广义上讲，PI计划执行以下操作：（1）开发一种Metrrized图的Jacobians理论，并在研究Berkovich分析曲线与其Jacobian之间的关系中应用； （2）将PI和Rum的潜在理论推广到伯科维奇投射线上，以更高的维度； （3）使用伯科维奇空间的理论在算术动力学中的一些开放问题上取得了新的进步。 在拟议的研究中要采用的方法涉及算术几何，分析，拓扑，图理论和动态系统的技术混合。 从广义上讲，该提案旨在解决数字理论中的关键统一原则，即应以对称方式对待全球领域的所有完成的想法。 近一百年来，这一直是数量理论的核心主题，例如，奇瓦利（Chevalley）对阶级田野理论的idelic表述和泰特（Tate）对adeles的谐波分析的发展。这项工作的主要知识优点是，它将导致伯克维奇（Berkovich）空间理论中的新发展，这是一个令人兴奋的主题，它已经对数字理论（包括lang langlands and norky e. norky symentions and the lang lang e. n angy symentions and the lang e e. necipry e。其他不同的领域。 该项目还将显示如何将伯科维奇的空间应用于与动态系统有关的各种有趣的问题。此外，PI将为Metrrized图理论带来新的想法。 Metrrized图（也在文献中也称为度量图或量子图）都在各种领域（包括物理学，数学生物学和数字理论）应用。 拟议工作的更广泛影响将包括分发研究和说明性论文，与不同领域的专家的互动以及对本科，研究生和博士后研究的支持。