Intersection Theory for Differential Equations

该项目由代数和数论项目和促进竞争研究的既定项目（EPSCoR）共同资助。该奖项涉及雅可比界猜想，这是微分代数中的一个基本问题，在各种数学学科中具有广泛的含义。首席研究员（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的推广工作延伸到YouTube等在线平台，在那里他们拥有一个拥有大量追随者的教育频道。此外，PI还通过演讲、视频和手稿向更广泛的受众传播复杂的数学概念，包括望月关于ABC猜想的工作。雅可比界猜想旨在确定描述任意非线性微分方程组通解所必需的积分常数的数目。该项目采用d -格式，变形理论，探讨了一般和退化的情况下，不同的设置，模堆栈，并应用于统一的朗威尔估计。该项目利用了微分代数方法的混合，包括微扰理论/∂-tan包和半连续性参数。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.