Intersection Theory for Differential Equations

该项目由代数和数论计划以及刺激竞争研究的既定计划（EPSCoR）共同资助。该奖项解决了Jacobi界猜想，这是微分代数中的一个基本问题，在各个数学学科中具有广泛的影响。 主要研究者（PI）积极参与教育工作，包括编写研究生水平的教科书和组织会议和研讨会。PI的外联工作延伸到YouTube等在线平台，在那里他们维持着一个拥有大量追随者的教育频道。此外，PI还通过讲座、视频、手稿等方式，向更多的人传播望月的ABC猜想等复杂的数学概念。Jacobi Bound猜想是为了确定任意非线性微分方程组的通解所需的积分常数的个数。该项目采用D-计划，变形理论，探索通用和退化的情况下，差异设置，模量堆栈，并应用于均匀Lang-Weil估计。该项目利用了微分代数方法的混合，包括微扰理论/正切束和半连续性参数。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR). This award addresses the Jacobi Bound Conjecture, a fundamental problem in differential algebra with broad implications across various mathematical disciplines. The principal investigator (PI) is actively engaged in educational efforts, including writing graduate-level textbooks and organizing conferences and seminars. The PI's outreach efforts extend to online platforms such as YouTube, where they maintain an educational channel with a substantial following. Moreover, the PI has been involved in disseminating complex mathematical concepts, including Mochizuki's work on the ABC Conjecture, to broader audiences through talks, videos, and manuscripts.The Jacobi Bound Conjecture seeks to determine the number of constants of integration necessary to describe a general solution for an arbitrary system of nonlinear differential equations. The project employs D-schemes, deformation theory, explores both generic and degenerate cases, the difference setting, moduli stacks, and applications to uniform Lang-Weil estimates. The project leverages a blend of differential algebraic methods, including perturbation theory/∂-tangent bundles and semi-continuity arguments.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.