Connections Between Number Theory, Algebraic Geometry, and Combinatorics

本提案关注算术几何、热带几何、Berkovich空间和组合学之间的新联系的发展。该建议的智力价值主要在于这些不同领域之间的相互作用，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还将表明，调和态射在微分几何和势理论中发挥着重要作用，在算术几何、热带几何和组合学中自然出现。应用程序将给出不同的主题阵列，包括Neron模型的组件组，热带交集理论和图论。最后，PI计划在Berkovich的解析空间理论和热带几何之间建立新的联系。这将使热带几何和Berkovich空间势理论的基础得到进一步发展，并将对最近关于热带椭圆曲线的一些结果提供更概念化的理解。提议工作的更广泛影响将包括应用于物理科学问题，与不同领域数学家的互动，以及对本科生和研究生研究的支持。例如，PI在图同构问题上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目前指导两名研究生，多年来一直积极参与本科生研究，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.