Connections Between Number Theory, Algebraic Geometry, and Combinatorics

这项建议涉及发展算术几何、热带几何、伯克维奇空间和组合学之间的新联系。这项建议的智力价值主要在于这些不同领域之间的相互影响，以及所提出的具体应用。例如，PI建议使用代数几何的思想来提供对图形同构问题的新见解，图形同构问题是图论和计算机科学中最著名的悬而未决的问题之一。PI还将展示调和态射，它在微分几何和位势理论中扮演着重要的角色，自然地出现在算术几何、热带几何和组合学中。课程将应用于一系列不同的学科，包括农隆模型的组成部分组、热带交叉点理论和图论。最后，PI计划在伯科维奇的解析空间理论和热带几何之间建立新的联系。这将有助于进一步发展热带几何的基础和Berkovich空间上的位势理论，并将提供对一些关于热带椭圆曲线的最新结果的更概念性的理解。拟议工作的更广泛影响将包括应用于物理科学中的问题，与不同领域的数学家的互动，以及对本科生和研究生研究的支持。例如，PI在图形同构问题上的新想法可能会在化学、生物学和计算机科学中得到应用。要实现本提案中提出的各种目标，PI将需要与数论、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这项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.