D-modules and invariants of singularities

奇点是几何对象以某种方式退化的点。它们在代数几何的研究中起着重要的作用。这个项目关注的是使用某些新的技术来研究奇点的不变量。这个理论提供了一个代数的方法来研究线性偏微分方程，但它已经发现了许多应用在代数几何和相关学科。这个项目的目标是，一方面，进一步研究一些最近引入的奇异类，另一方面，探索与zeta函数和不变量的新联系。该提案为PI提供了与研究生和博士后合作的充分机会。他也将写一本关于这个主题的研究专著。这个项目将涉及几个不同的研究方向。首先，PI将扩展k-有理和k-Du Bois奇点的研究，超越超曲面的情况，使用最近定义的局部完全相交的最小指数。在另一个方向，他将研究一个给定素数的奇异模幂的D-模理论不变量，从正特征中提炼测试理想和F-纯阈值，并可能提供一个算术对应的特征零不变量，如霍奇理想和最小指数。第三个方向是探索动机zeta函数和最小指数之间的联系，以及拓扑zeta函数和D-模之间的联系（可能应用于强单值性猜想）。该奖项反映了NSF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。

Singularities are points where a geometric object degenerates in some way. They play an important role in the study of algebraic geometry. This project is concerned with the use of certain novel techniques to study invariants of singularities. This theory provides an algebraic approach to the study of linear partial differential equations, but it has found many applications in algebraic geometry and related subjects. The goal of this project is, on one hand, to further study some recently introduced classes of singularities and, on the other hand, to explore new connections with zeta functions and invariants. The proposal provides ample opportunity for the PI to collaborate with graduate students and post-docs. He will also write a research monograph on this subject.This project will involve several different research directions. First, the PI will expand the study of k-rational and k-Du Bois singularities beyond the case of hypersurfaces, using the recently defined version of minimal exponent for locally complete intersections. In a different direction, he will investigate D-module theoretic invariants of singularities modulo powers of a given prime, that refine the test ideals and F-pure thresholds from positive characteristic, and that might provide an arithmetic counterpart to characteristic zero invariants, such as Hodge ideals and minimal exponents. A third direction concerns exploring connections between the motivic zeta function and the minimal exponent on one side, and between the topological zeta function and D-modules on the other side (with possible applications to the Strong Monodromy Conjecture).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.