Non-Archimedean Methods in Analysis, Dynamics and Geometry

马蒂亚斯·琼森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，以研究分析、动力学和几何中的一系列问题。Jonsson将研究Berkovich空间，这是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，一个相当于证明了关于多次调和函数的四个不同的等价性概念是等价的。这是复杂分析中的一个问题，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。给出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将研究算术复杂性是如何沿着动力学轨道增长的。在这里，这个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。琼森还将彻底学习非阿基米德分析和动力学。例如，他的目标是推广与他的合作者Bouocksom和Favre共同获得的关于非阿基米德Monge-Ampere方程的成功分析。Mattias Jonsson的这个数学研究项目是在分析、几何和动力学领域进行的，自从他们发现以来，这些领域一直被用来理解我们周围的世界。它们对其他科学领域至关重要，如工程学、生物学和经济学。例如，对任何随时间发生变化的现象(如体内细菌数量、股票市场等)进行数学建模。可以看作是一个动力系统。同样，几何学是当前许多工业应用(如3D打印)的基础。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，根据这些公理或基本假设，可以从逻辑上推导出所有其他合理的性质。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有效时发生的某些现象。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。

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.