Berkovich Spaces, Tropical Geometry, Combinatorics, and Dynamics

这个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方面有重要的应用。 这项工作还涉及与图论的联系，该研究领域可应用于互联网等大规模网络的研究。 该项目将产生一些更广泛的影响，包括支持高中，本科和研究生的研究。 PI目前正在监督两名博士。格鲁吉亚理工学院的学生和一名博士除了一名本科研究生和两名高中生之外，他们每个人都在从事与这个项目有关的项目。 PI还参与了一些数学推广活动，包括为天才高中生开设的数论和密码学在线课程，他还写了一个受欢迎的数学博客。 这些活动将在本项目中继续进行，并在可行时纳入当前研究的想法。 该项目还将通过组织会议和出版会议记录，促进数学思想的传播。 该项目的主要智力价值是，它将增加我们对Berkovich空间，热带几何，组合数学和复杂动力学的理解，并挖掘这些不同研究领域之间的新关系。 伯科维奇空间是二十世纪早期库尔特·亨泽尔关于“p-adic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都有应用，包括代数几何、数论和复动力学（它们可以用来研究分形，如曼德尔布罗特集）。 伯科维奇空间也与热带几何密切相关，热带几何是一个相对较新且非常活跃的研究领域，在数论、代数几何、统计学、生物学和物理学中均有应用。 人们可以把热带几何看作是一个简化的经典代数几何模型，其中多项式方程组的公共解被一个更简单的线性不等式系统的公共解所取代。 令人惊讶的是--而且相当神秘的是--热带简化记住了比人们最初预期的更多的关于原始解决方案的信息。 研究人员以前的工作涉及布科维奇空间和热带几何的意想不到的新应用，这个项目将建立在并大大推进这项工作。

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.