Connections Between Number Theory, Algebraic Geometry, and Combinatorics

数论、代数几何和组合数学之间的联系

• 批准号: 0901487
• 负责人: Matthew Baker
• 金额: $ 30.07万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901487&HistoricalAwards=false
• 关键词: Connections; Between; Number; Theory; Algebraic

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.

该提案涉及算术几何、热带几何、贝尔科维奇空间和组合学之间新联系的发展。该提案的智力价值主要在于这些不同领域之间的交叉融合，以及所提出的具体应用。 例如，PI建议利用代数几何的思想为图同构问题提供新的见解，这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论中发挥着重要作用，自然出现在算术几何、热带几何和组合数学中。 应用程序将应用于各种学科，包括 Neron 模型的组成部分、热带交叉理论和图论。 最后，PI 计划在贝尔科维奇的解析空间理论和热带几何之间建立新的联系。 这将使热带几何和伯科维奇空间势理论的基础得到进一步发展，并且还将为有关热带椭圆曲线的一些最新结果提供更概念性的理解。拟议工作的更广泛影响将包括在物理科学问题中的应用、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 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。 实现该提案中提出的各种目标需要 PI 与数论、代数几何、组合学和动力系统领域的领先专家进行互动。 该项目负责人目前正在指导两名研究生，多年来一直积极参与本科生研究，他计划与各级学生合作开展与该提案相关的研究项目。