Non-Archimedean Methods in Analysis, Dynamics and Geometry

Mattias Jonsson的这个数学研究项目将使用非阿基米德分析和几何的技术来研究分析、动力学和几何中的一系列问题。Jonsson将研究Berkovich空间，实流形和复流形的非阿基米德类似物。在许多具体的项目中，一个等于证明了四种不同的多次谐波函数的等奇异性概念是等价的。这是一个复杂分析中的问题，但它的解决方法涉及在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，Jonsson将研究算术复杂度如何沿着动力学轨道增长。这里的问题是用相当初级的数论来表述的，但Jonsson将通过对Berkovich空间上的诱导动力系统的详细研究来解决这个问题。琼森还将直接研究非阿基米德分析和动力学。例如，他的目标是扩展与他的合作者Boucksom和Favre共同获得的关于非阿基米德蒙日-安培方程的成功分析。Mattias Jonsson的这个数学研究项目涉及分析、几何和动力学领域，自从它们被发现以来，就一直被用来理解我们周围的世界。它们对工程学、生物学和经济学等其他科学领域也至关重要。例如，任何随时间变化的现象（如人体中的细菌数量、股票市场等）的数学模型都可以看作是一个动态系统。同样，几何是当前许多工业应用（如3D打印）的基础。这个项目将增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，所有其他合理的性质都可以从这些公理或基本假设中逻辑地推导出来。这个项目的主要焦点是详细研究当阿基米德公理（阿基米德的Syracuse）不再有效时发生的某些现象。结果证明，即使主要研究对象是通常的阿基米德式的数学，其结果也是有用的。

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.