Tropical and Non-Archimedean Analytic Methods in Algebraic Geometry

代数几何研究多项式方程（代数变体）系统的解集，在数学及其他领域有许多应用，包括物理、计算机科学和工程。该奖项支持的研究将集中于利用现代退化技术，特别是热带几何领域的退化技术，从代数几何中研究经典空间。热带几何的主要目标是将代数变异问题转化为多面体复合体问题。一种称为热带化的过程将多面体复合体附着在代数变体上。多面体复合体，一个组合对象，编码了一些原始代数变化的几何。该研究开发了进一步的工具，以研究代数变种的所谓的“非阿基米德分析”。总的来说，这个项目将提炼、抽象和推广这些新方法，并探索代数几何中开放问题的更深层次应用。与此研究项目相结合，PI将继续指导耶鲁大学本科数学研究SUMRY项目，不仅为本科生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将扩展在证明Gieseker-Petri定理和二次多项式的极大秩猜想中发展起来的热带无关方法，发展线性序列的热带化的新概念，并将分段线性方法与有理函数约简相结合，在极大秩猜想和强极大秩猜想方面取得进一步进展。他将继续他的工作，利用稳定热带曲线模空间的组合拓扑来提取曲线模空间的顶权上同调的信息，寻找额外的结构，如隐藏过滤，其梯度块相对于标记点置换引起的作用是稳定的。PI还旨在通过将代数曲线上连接的矢量束的收敛牛顿多边形理论推广到高维变量上具有可积连接的矢量束，以环向矢量束作为测试案例，为矢量束的热带理论奠定基础。

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.