Berkovich Spaces, Tropical Geometry, Combinatorics, and Dynamics

这项研究项目主要涉及数论方面的工作，数论是数学的一个分支，在密码学和编码理论中有重要的应用。这项工作还包括与图论的联系，图论是一个研究领域，应用于研究大规模网络，如互联网。该项目将产生许多更广泛的影响，包括对高中、本科和研究生研究的支持。PI目前正在指导两名来自佐治亚理工学院的博士生和一名来自加州大学伯克利分校的博士生，另外还有一名本科生和两名高中生，他们每个人都在从事与这个项目相关的项目。PI还参与了许多数学推广活动，包括为有天赋的高中生开设的数论和密码学在线课程，他还写了一个很受欢迎的数学博客。这些活动将在这个项目中继续进行，只要可行，就会整合当前研究的想法。该项目还将通过组织会议和出版会议记录，为传播数学思想作出贡献。该项目的主要学术价值是，它将增加我们对伯克维奇空间、热带几何、组合学和复杂动力学的理解，并揭示这些不同研究领域之间的新关系。Berkovich空间--Kurt Hensel在二十世纪早期关于“p-ady数”的开创性工作的现代化身--近年来在数学研究的许多领域得到了应用，包括代数几何、数论和复杂动力学(在这些领域中，它们可以用来研究诸如Mandelbrot集的分形学)。Berkovich空间还与热带几何密切相关，热带几何是一个相对较新和非常活跃的研究领域，应用于数论、代数几何、统计学、生物学和物理学。人们可以把热带几何看作是一个简化的模型经典代数几何，其中多项式方程组的公共解集被一个简单得多的线性不等式组的公共解集所代替。令人惊讶的是--也是相当神秘的--热带简化方法记住的关于原始解决方案的信息比人们最初预期的要多得多。这位研究人员之前的工作涉及到伯克维奇空间和热带几何的意想不到的新应用，而这个项目将在这一工作的基础上继续并显著推进这一工作。

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.