Tropical and Non-Archimedean Analytic Methods in Algebraic Geometry

代数几何研究多项式方程组（代数簇）的解集，在数学和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在使用现代退化技术，特别是来自热带几何领域的技术，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的问题转化为多面体复形的问题。一个称为热带化的过程将一个多面体复形附加到一个代数簇上。多面体复形是一个组合对象，它编码了原始代数簇的一些几何。该研究为代数簇的“非阿基米德分析”研究提供了进一步的工具。总的来说，这个项目将提炼，抽象和概括这些新方法，并探索更深层次的应用，以解决代数几何中的开放问题。与此研究计划一起，PI将继续指导耶鲁大学数学本科研究的SUMRY计划，不仅为本科参与者提供研究机会，还为研究生和博士后领导小组项目提供指导机会。PI将扩展在Gieseker-Petri定理和二次型最大秩猜想的证明中开发的热带独立方法，开发线性级数热带化的新概念，并将分段线性方法与有理函数的简化相结合，以进一步推进最大秩猜想和强最大秩猜想。他将继续他的工作，使用组合拓扑的模空间的稳定热带曲线，以提取信息的最高权重上同调模空间的曲线，寻找额外的结构，如隐藏的filtrations的分级件是代表稳定的相对于行动引起的置换的标记点。PI还旨在为向量丛的热带理论奠定基础，通过将代数曲线上连接的向量丛的收敛牛顿多边形理论推广到高维簇上具有可积连接的向量丛，使用环面向量丛作为测试案例。

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.