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

复杂分析、动力学和几何中的非阿基米德方法

• 批准号: 1900025
• 负责人: Mattias Jonsson
• 金额: $ 24万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900025&HistoricalAwards=false
• 关键词: Non; Archimedean; Methods; Complex; Analysis

Analysis, Geometry and Dynamics are areas of Mathematics, which 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 research 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a detailed study of certain phenomena that occur when the Archimedean Axiom, attributed to Archimedes of Syracuse, is no longer valid.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.This mathematics research project will employ methods from non-Archimedean analysis and geometry in order to study a range of problems in analysis, dynamics and geometry. The principal investigator will work with Berkovich spaces, non-Archimedean analogues of real and complex manifolds. Among many specific projects, one involves a detailed study of the space of metrics on an ample line bundle on a compact complex manifold. Geodesic rays inside this space can be studied using non-Archimedean means. The principal investigator will also continue his work with Boucksom on the Kontsevich-Soibelman conjecture that originally arose in the study of mirror symmetry. In analysis and dynamics, the principal investigator will try to use non-Archimedean techniques to construct invariant currents for quite general two-dimensional complex dynamical systems and study the growth of arithmetic complexity along orbits of certain 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.

分析，几何和动力学是数学的领域，自从它们被发现以来，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。该研究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，即使当主要研究对象是通常的，阿基米德，kind.这个数学研究项目将采用非阿基米德分析和几何的方法，以研究一系列的问题，在分析，动力学和几何。主要研究者将与伯科维奇空间，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，一个涉及到一个紧凑的复杂流形上的充足线丛的度量空间的详细研究。在这个空间内的测地线可以使用非阿基米德方法来研究。首席研究员还将继续他的工作与Boucksom的Kontsevich-Soibelman猜想，最初出现在研究镜像对称。在分析和动力学方面，主要研究者将尝试使用非阿基米德技术为相当一般的二维复杂动力系统构建不变流，并研究算术复杂性沿着某些算术动力系统的轨道的增长。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值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

Erratum to “Uniform K-stability and asymptotics of energy functionals in Kähler geometry”

“凯勒几何中能量泛函的均匀 K 稳定性和渐近性”勘误表

• DOI: 10.4171/jems/1215
• 发表时间: 2022
• 期刊: Journal of the European Mathematical Society
• 影响因子: 2.6
• 作者: Boucksom, Sébastien;Hisamoto, Tomoyuki;Jonsson, Mattias
• 通讯作者: Jonsson, Mattias

A transcendental dynamical degree

超越的动力程度

• DOI: 10.4310/acta.2020.v225.n2.a1
• 发表时间: 2020
• 期刊: Acta Mathematica
• 影响因子: 3.7
• 作者: Bell, Jason P.;Diller, Jeffrey;Jonsson, Mattias
• 通讯作者: Jonsson, Mattias

Convergence of $p$-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces

伯科维奇空间中的 $p$-adic 多规范测量与勒贝格测量的收敛

• DOI: 10.5802/jep.118
• 发表时间: 2020
• 期刊: Journal de l’École polytechnique — Mathématiques
• 作者: Jonsson, Mattias;Nicaise, Johannes
• 通讯作者: Nicaise, Johannes

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

Thresholds, valuations, and K-stability

• DOI: 10.1016/j.aim.2020.107062
• 发表时间: 2017-06
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Harold Blum;Mattias Jonsson