Non-Archimedean Techniques in Analysis, Dynamics, and Geometry

这个数学研究项目涉及分析、几何和动力学领域，这些领域是数学的核心，也是其他科学领域的关键，如工程、生物和经济学。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。这个项目将加强这些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从逻辑上推导出其他合理的性质。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。这项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。该项目利用了Berkovich空间，即实流形和复流形的非阿基米德类似物。该项目的一部分涉及Calabi-Yau流形，即在数学物理中发挥重要作用的几何对象。主要的研究人员和合作者将使用非阿基米德技巧来研究关于退化为实的Calabi-Yau流形的单参数族的猜想。该项目的另一个部分是算术动力学。给出了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，研究了算法复杂性是如何沿着动力学轨道增长的。在这里，这个问题是用初等数论表述的，但通过对Berkovich空间上的诱导动力系统的详细研究来解决。

This mathematics research project concerns the areas of analysis, geometry, and dynamics, which are central to mathematics and 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 these branches of mathematics. The development of geometry goes back to the ancient Greeks, who laid down axioms, or basic assumptions, from which 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a detailed study of certain phenomena that occur when the Archimedean axiom, attributed to Archimedes of Syracuse, is no longer valid.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 research project will use techniques from non-Archimedean analysis and geometry in order to study a range of problems in analysis, dynamics, and geometry. The project makes use of Berkovich spaces, non-Archimedean analogues of real and complex manifolds. One part of the project involves Calabi-Yau manifolds, geometric objects that play an important role in mathematical physics. The principal investigator and collaborator will use non-Archimedean techniques to study a conjecture concerning one-parameter families of complex Calabi-Yau manifolds that degenerate to a real Calabi-Yau manifold. Another part of the project is in arithmetic dynamics. Given a discrete-time two-dimensional dynamical system defined by a pair of polynomials with rational coefficients, the research investigates how the arithmetic complexity grows along orbits of the dynamics. Here the problem is formulated in terms of elementary number theory but approached using a detailed study of an induced dynamical system on a Berkovich space.