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

这个项目调查分析，几何和动力系统领域的开放性问题。这些数学学科对许多科学领域都至关重要，包括工程学、生物学和经济学。例如，任何随时间变化的物理现象的数学建模都可以被视为一个动态系统。同样，几何形状是当前许多工业应用（如3D打印）的基础。这一研究项目将加强上述数学领域的现有工具。几何学的公理化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假设中逻辑推导出来的。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被假设时发生的某些现象。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。该项目还将产生各种层次的研究机会，适合研究生和本科生的工作。该项目将采用非阿基米德工具来研究分析，动力学和几何学中的一系列问题。这一奋进的关键是对伯科维奇空间的理解，它是真实的和复流形的非阿基米德类似物。该项目的一个组成部分涉及到一个紧凑的复杂流形上的充足的线丛度量空间的详细研究。这个空间中的测地线可以通过非阿基米德方法来研究。另一个需要考虑的主题是孔采维奇-索贝尔曼猜想，它最初出现在镜像对称的研究中。在分析和动力学领域，将采用非阿基米德技术来构建二维复杂动力系统的不变流，从而研究算术动力系统的算术复杂性沿着轨道的增长。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。

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.