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

分析、几何和动力学是数学的领域，自从它们被发现以来，就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。这一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，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。这个数学研究项目将使用非阿基米德分析和几何的方法来研究分析、动力学和几何中的一系列问题。首席研究员将研究Berkovich空间，即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，一个涉及到紧致复流形上的充裕线丛上的度量空间的详细研究。这个空间内的测地线可以用非阿基米德平均来研究。首席研究员还将继续与布克索姆就康采维奇-索贝尔曼猜想开展工作，该猜想最初出现在镜面对称性的研究中。在分析和动力学方面，首席研究员将尝试使用非阿基米德技术来构造相当一般的二维复杂动力系统的不变流，并研究某些算术动力系统沿轨道的算术复杂性的增长。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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a detailed study of certain phenomena that occur when the Archimedean Axiom, attributed to Archimedes of Syracuse, is no longer valid.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.