Intersection Theory for Differential Equations

该项目由代数和数论项目和已建立的激励竞争性研究项目(EPSCoR)共同资助。该奖项涉及雅可比界限猜想，这是微分代数中的一个基本问题，具有广泛的跨数学学科的意义。首席调查员(PI)积极参与教育工作，包括编写研究生水平的教科书以及组织会议和研讨会。PI的外展努力延伸到YouTube等在线平台，在这些平台上，他们保持着一个拥有大量追随者的教育频道。此外，PI还参与了通过演讲、视频和手稿向更广泛的受众传播复杂的数学概念，包括Mochizuki关于ABC猜想的工作。雅可比界限猜想寻求确定描述任意非线性微分方程组的一般解所需的积分常数的数目。该项目使用D-格式，形变理论，探索一般和退化情况，差分设置，模堆叠，以及在一致Lang-Weil估计中的应用。该项目利用了微分代数方法的混合，包括微扰理论/∂-切丛和半连续论证。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智力优势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.