Complex Analysis, Dynamics, and Geometry via Non-Archimedean Methods

通过非阿基米德方法进行复杂分析、动力学和几何

• 批准号: 2154380
• 负责人: Mattias Jonsson
• 金额: $ 42.61万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154380&HistoricalAwards=false
• 关键词: Complex; Analysis; Dynamics; Geometry; via

This project investigates open questions in the areas of analysis, geometry, and dynamical systems. These mathematical disciplines are crucial for a host of scientific fields, including engineering, biology, and economics. For example, the mathematical modeling of any physical phenomenon that undergoes change over time can be viewed as a dynamical system. Similarly, geometry is the basis for numerous current industrial applications such as 3D printing. This research project will enhance the tools available in the aforementioned mathematical areas. The axiomatic development of geometry, which postulates a set of basic assumptions from which all other reasonable conclusions are logically deduced, dates back to ancient Greece. The major focus of this project is a detailed study of certain phenomena that occur when the Archimedean axiom, attributed to Archimedes of Syracuse, is no longer postulated. The resulting mathematics turns out to be useful even when the primary object of study is of the usual, Archimedean, kind. The project will also generate research opportunities at a variety of levels, suitable for work by graduate and undergraduate students.This project will employ non-Archimedean tools to study a range of problems in analysis, dynamics, and geometry. Essential to this endeavor is an understanding of Berkovich spaces, which are non-Archimedean analogues of real and complex manifolds. One component of the project involves a detailed study of the space of metrics on an ample line bundle on a compact complex manifold. Geodesic rays within this space can be studied via non-Archimedean methods. Another topic for consideration is the Kontsevich-Soibelman conjecture, which originally arose in the study of mirror symmetry. In the areas of analysis and dynamics, non-Archimedean techniques will be employed to construct invariant currents for two-dimensional complex dynamical systems, allowing for a study of the growth of arithmetic complexity along orbits of arithmetic dynamical systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

本项目研究分析、几何和动力系统领域的公开问题。这些数学学科对包括工程学、生物学和经济学在内的许多科学领域至关重要。例如，对任何随时间发生变化的物理现象的数学建模可以被视为一个动力系统。同样，几何学是当前许多工业应用(如3D打印)的基础。这一研究项目将加强上述数学领域的现有工具。几何学的公理发展可以追溯到古希腊，它假定了一系列基本假设，所有其他合理的结论都是从这些基本假设中逻辑推导出来的。这个项目的主要焦点是详细研究当被认为是锡拉丘兹的阿基米德的阿基米德公理不再被假设时发生的某些现象。结果证明，即使研究的主要对象是通常的阿基米德式的数学，结果也是有用的。该项目还将产生不同层次的研究机会，适合研究生和本科生的工作。该项目将使用非阿基米德工具来研究分析、动力学和几何中的一系列问题。这一努力的关键是理解Berkovich空间，它是实流形和复流形的非阿基米德类似。该项目的一个组成部分涉及详细研究紧致复杂流形上的充裕线丛上的度量空间。该空间内的测地线可通过非阿基米德方法进行研究。另一个要考虑的话题是康采维奇-索贝尔曼猜想，它最初出现在镜面对称性的研究中。在分析和动力学领域，非阿基米德技术将被用于构造二维复杂动力系统的不变流，从而允许研究算术动力系统沿轨道的算术复杂性的增长。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Global pluripotential theory over a trivially valued field

• DOI: 10.5802/afst.1705
• 发表时间: 2022-07
• 期刊: Annales de la Faculté des sciences de Toulouse : Mathématiques
• 作者: S. Boucksom;Mattias Jonsson