Tropical and Non-Archimedean Analytic Methods in Algebraic Geometry

代数几何中的热带和非阿基米德解析方法

• 批准号: 1702428
• 负责人: Sam Payne
• 金额: $ 24万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-09-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1702428&HistoricalAwards=false
• 关键词: Tropical; Non; Archimedean; Analytic; Methods

Algebraic geometry studies solution sets of systems of polynomial equations (algebraic varieties) and has many applications in areas within mathematics and beyond, including physics, computer science, and engineering. The research supported by this award will center on using modern degeneration techniques, especially those from the field of tropical geometry, to study classical spaces from algebraic geometry. The main goal of tropical geometry is transforming questions about algebraic varieties into questions about polyhedral complexes. A process called tropicalization attaches a polyhedral complex to an algebraic variety. The polyhedral complex, a combinatorial object, encodes some of the geometry of the original algebraic variety. The research develops further tools for the study of algebraic varieties in terms of their so called "nonarchimedean analytification". Overall this project will refine, abstract, and generalize these new methods and explore deeper applications to open problems in algebraic geometry. In conjunction with this research program, the PI will continue to direct the SUMRY program for undergraduate research in mathematics at Yale, providing not only research opportunities for the undergraduate participants, but also mentorship opportunities for the graduate students and postdocs leading small group projects. The PI will extend the tropical independence methods developed in proofs of the Gieseker-Petri theorem and the maximal rank conjecture for quadrics, developing new notions of tropicalization of linear series and combining piecewise linear methods with reduction of rational functions to pursue further progress toward the maximal rank conjecture and strong maximal rank conjecture. He will continue his work using the combinatorial topology of moduli spaces of stable tropical curves to extract information about the top weight cohomology of moduli spaces of curves, looking for additional structures such as hidden filtrations whose graded pieces are representation stable with respect to the action induced by permutation of the marked points. The PI also aims to develop foundations for a tropical theory of vector bundles, by generalizing the theory of convergence Newton polygons for vector bundles with connection on algebraic curves to vector bundles with integrable connections on higher dimensional varieties, using toric vector bundles as a test case.  

代数几何研究多项式方程组（代数簇）的解集，在数学及其他领域有许多应用，包括物理学、计算机科学和工程学。该奖项支持的研究将集中于使用现代退化技术，特别是那些技术。从热带几何领域，从代数几何研究经典空间 热带几何的主要目标是将代数簇问题转化为多面体复形问题，这种过程称为热带化。多面体复形是一种组合对象，它编码了原始代数簇的一些几何形状，该研究在所谓的“非阿基米德分析”方面开发了进一步的工具。 、抽象和概括这些新方法，并探索代数几何中开放问题的更深入应用，PI 将继续指导 SUMRY 项目。耶鲁大学数学本科生研究，不仅为本科生参与者提供研究机会，还为领导小组项目的研究生和博士后提供指导机会。PI 将扩展在吉塞克-佩特里定理的证明中开发的热带独立方法。二次函数的最大秩猜想，发展线性级数热带化的新概念，并将分段线性方法与有理函数约简相结合，以进一步推进最大秩猜想和强最大秩他将继续使用稳定热带曲线模空间的组合拓扑来提取有关曲线模空间的顶部权重上同调的信息，寻找其他结构，例如其分级部分相对于表示稳定的隐藏过滤。 PI 还旨在通过推广牛顿收敛理论，为矢量丛热带理论奠定基础。使用环面向量丛作为测试用例，将具有代数曲线连接的向量丛的多边形转换为具有更高维度品种上可积连接的向量丛。