Intersection Theory for Differential Equations

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.