Connections Between Number Theory, Algebraic Geometry, and Combinatorics

该建议关注算术几何、热带几何、布氏空间和组合学之间的新联系的发展，其智力价值主要在于这些不同领域之间的相互促进，以及所提出的具体应用。 例如，PI建议使用来自代数几何的思想来提供对图同构问题的新见解，图同构问题是图论和计算机科学中最著名的未解决问题之一。PI还将展示调和态射，它在微分几何和势理论中起着突出的作用，在算术几何，热带几何和组合学中自然出现。 应用程序将被赋予一个不同的主题，包括组件组的Neron模型，热带交叉理论和图论。 最后，PI计划开发新的连接之间的伯科维奇的理论分析空间和热带几何。 这将使进一步发展的基础上的热带几何和潜在的理论Berkovich空间，也将提供一个更概念性的理解最近的一些结果有关热带椭圆curves.The更广泛的影响，拟议的工作将包括应用到物理科学中的问题，与数学家在不同领域的互动，并支持本科生和研究生的研究。 例如，PI关于图同构问题的新想法可能会应用于化学，生物学和计算机科学。 要实现本提案中提出的各种目标，PI需要与数论、代数几何、组合学和动力系统领域的顶尖专家进行互动。 PI目前正在指导两名研究生，多年来一直积极参与本科研究，计划与各级学生合作开展与该提案相关的研究项目。

This proposal is concerned with the development of new connections between arithmetic geometry, tropical geometry, Berkovich spaces, and combinatorics.The intellectual merit of the proposal lies primarily in the cross-fertilization between these different areas, and in the concrete applications being proposed. For example, the PI proposes to use ideas coming from algebraic geometry to provide new insight into the graph isomorphism problem, one of the most famous unsolved problems in graph theory and computer science.The PI will also show that harmonic morphisms, which play a prominent role in differential geometry and potential theory, arise naturally in arithmetic geometry, tropical geometry, and combinatorics. Applications will be given to a diverse array of subjects including component groups of Neron models, tropical intersection theory, and graph theory. Finally, the PI plans to develop new connections between Berkovich's theory of analytic spaces and tropical geometry. This will enable further development of the foundations of tropical geometry and potential theory on Berkovich spaces, and will also provide a more conceptual understanding of some recent results concerning tropical elliptic curves.The broader impacts of the proposed work will include applications to problems in the physical sciences, interaction with mathematicians in different fields, and support for undergraduate and graduate research. For example, the PI's new ideas on the graph isomorphism problem could potentially have applications to chemistry, biology, and computer science. Accomplishing the various goals laid out in this proposal will require the PI to interact with leading experts in the fields of number theory, algebraic geometry, combinatorics, and dynamical systems. The PI, who is currently supervising two graduate students and has been intensely involved for many years with undergraduate research, plans to work with students at all levels on research projects related to this proposal.