Non-Archimedean Methods in Analysis, Dynamics and Geometry

分析、动力学和几何中的非阿基米德方法

• 批准号: 1266207
• 负责人: Mattias Jonsson
• 金额: $ 32.5万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1266207&HistoricalAwards=false
• 关键词: Non; Archimedean; Methods; Analysis; Dynamics

This mathematics research project by Mattias Jonsson will use techniques from non-Archimedean analysis and geometry in order to study a range of problems in analysis, dynamics and geometry. Jonsson will work with Berkovich spaces, non-Archimedean analogues of real and complex manifolds. Among many specific projects, one amounts to showing that four different notions of equisingularity for plurisubharmonic functions are equivalent. This is a problem in complex analysis but its solution involves doing analysis on a Berkovich space. Another project is in arithmetic dynamics. Given a discrete-time two-dimensional dynamical system defined by a pair of polynomials with rational coefficients, Jonsson will investigate how the arithmetic complexity grows along orbits of the dynamics. Here the problem is formulated using rather elementary number theory but Jonsson will approach it using a detailed study of an induced dynamical system on a Berkovich space. Jonsson will also study non-Archimedean analysis and dynamics outright. For example, he aims to extend the successful analysis obtained jointly with his collaborators Boucksom and Favre on the non-Archimedean Monge-Ampere equation.This mathematics research project by Mattias Jonsson is in the areas of Analysis, Geometry and Dynamics which, ever since their discovery have been used in order to understand the world around us. They are crucial to other scientific fields, such as engineering, biology and economics. For example, the mathematical modeling of any phenomenon that undergoes change over time (such as the population of bacteria in a body, the stock market, etc.) can be viewed as a dynamical system. Similarly, geometry is the basis for many current industrial applications such as 3D printing. This project will enhance the tools available in the aforementioned branches of mathematics. The development of geometry goes back to the ancient Greeks, who laid down axioms, or basic assumptions, from which all other reasonable properties could be logically deduced. The main 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 valid. The resulting mathematics turns out to be useful even when the primary object of study is of the usual, Archimedean, kind.

Mattias Jonsson的这个数学研究项目将使用非阿基米德分析和几何技术，以研究分析，动力学和几何学中的一系列问题。琼森将与伯科维奇空间，非阿基米德类似物的真实的和复杂的流形。在许多具体的项目中，有一个项目表明，四种不同的多重次调和函数的等奇异性概念是等价的。这是复分析中的一个问题，但它的解决方案涉及到在Berkovich空间上进行分析。另一个项目是算术动力学。给定一个由一对有理系数多项式定义的离散时间二维动力系统，琼森将研究算术复杂性如何沿着动力学的轨道沿着增长。这里的问题是制定使用而初等数论，但琼森将接近它使用一个详细的研究诱导动力系统的伯科维奇空间。琼森还将研究非阿基米德分析和动力学彻底。例如，他的目标是扩展与他的合作者Boucksom和Favre在非阿基米德Monge-Ampere方程上共同获得的成功分析。Mattias Jonsson的这个数学研究项目是在分析，几何和动力学领域，自从他们发现以来一直被用来理解我们周围的世界。它们对其他科学领域，如工程学、生物学和经济学至关重要。例如，任何现象的数学建模，经历随着时间的变化（如细菌在体内的种群，股票市场等）。可以看作是一个动态系统。同样，几何形状是当前许多工业应用（如3D打印）的基础。该项目将加强上述数学分支中可用的工具。几何学的发展可以追溯到古希腊人，他们制定了公理或基本假设，从这些公理或基本假设中可以逻辑地推导出所有其他合理的性质。这个项目的主要重点是详细研究当锡拉丘兹的阿基米德公理不再有效时发生的某些现象。由此产生的数学证明是有用的，即使主要的研究对象是通常的，阿基米德，种类。