Berkovich Spaces, Tropical Geometry, Combinatorics, and Dynamics

伯科维奇空间、热带几何、组合学和动力学

• 批准号: 1502180
• 负责人: Matthew Baker
• 金额: $ 24万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1502180&HistoricalAwards=false
• 关键词: Berkovich; Spaces; Tropical; Geometry; Combinatorics

This research project is primarily concerned with work in number theory, a branch of mathematics with important applications to cryptography and coding theory. The work also features connections to graph theory, a research area with applications to the study of large-scale networks such as the internet. The project will have a number of broader impacts, including support for high school, undergraduate, and graduate research. The PI is currently supervising two Ph.D. students from Georgia Tech and one Ph.D. student from UC Berkeley, in addition to one undergraduate research student and two high school students, each of whom is working on projects related to this project. The PI also has been involved in a number of mathematical outreach activities, including an online course on Number Theory and Cryptography for gifted high school students, and he writes a popular math blog. These activities will be continued in this project, integrating ideas from the current research whenever feasible. The project will also contribute to the dissemination of mathematical ideas through conference organization and publication of proceedings. The primary intellectual merit of the project is that it will increase our understanding of Berkovich spaces, tropical geometry, combinatorics, and complex dynamics, and unearth new relationships between these different areas of research. Berkovich spaces -- the modern incarnation of the pioneering early twentieth century work of Kurt Hensel on "p-adic numbers" -- have in recent years found applications to numerous areas of mathematical research, including algebraic geometry, number theory, and complex dynamics (where they can be used to study fractals such as the Mandelbrot set). Berkovich spaces are also intimately connected with tropical geometry, a relatively new and very active area of research with applications to number theory, algebraic geometry, statistics, biology, and physics. One can think of tropical geometry as a simplified model classical algebraic geometry in which the set of common solutions to a system of polynomial equations is replaced by the set of common solutions to a much simpler system of linear inequalities. Surprisingly -- and rather mysteriously -- the tropical simplification remembers much more information about the original solutions than one might originally expect. The investigator's previous work involved unexpected new applications of Berkovich spaces and tropical geometry, and this project will build on and significantly advance that work.

该研究项目主要涉及数字理论的工作，这是数学的一个分支，具有重要应用于密码学和编码理论。 这项工作还具有与Graph Doemon的连接，该研究领域具有针对大型网络（例如Internet）的研究。 该项目将产生更广泛的影响，包括对高中，本科和研究生研究的支持。 PI目前正在监督两个博士学位。佐治亚理工学院的学生和一位博士学位来自加州大学伯克利分校的学生除了一名大学生和两名高中生，每个学生都在从事与该项目有关的项目。 PI还参与了许多数学外展活动，包括针对有天赋的高中生编号理论和密码学的在线课程，他写了一个受欢迎的数学博客。 这些项目将在该项目中继续进行，并在可行的情况下整合当前研究的想法。 该项目还将通过会议组织和诉讼发布来促进数学思想的传播。 该项目的主要智力优点是，它将提高我们对伯科维奇空间，热带几何形状，组合动力学和复杂动态的理解，以及这些不同研究领域之间的新关系。 伯科维奇空间 - 二十世纪初期的库尔特·汉塞尔（Kurt Hensel）在“ p-adic数字”上的现代化身 - 近年来发现了许多数学研究领域的应用，包括代数几何，数字理论和复杂的动力学（复杂的动力学）（它们可用于研究分形等分形，例如Mandelbrot集）。 Berkovich空间还与热带几何形状密切相关，这是一个相对较新且非常活跃的研究领域，应用了数字理论，代数几何，统计，生物学和物理学。 人们可以将热带几何形状视为一种简化的模型经典代数几何形状，其中多项式方程系统的一组通用解决方案被一组通用解决方案所取代，以实现更简单的线性不等式系统。 令人惊讶的是 - 而且很神秘 - 热带简化回忆起有关原始解决方案的信息，比人们最初预期的要多。 研究者以前的工作涉及伯科维奇空间和热带几何形状的意外新应用，该项目将基于并大大推动这项工作。