CAREER: Local Cohomology, de Rham Cohomology and D-modules

提议的项目是在交换代数领域，这是一个代数（多项式）方程组的解在交换环的研究。交换环可以看作是整数的抽象类比，其中可以加、减、乘元素。在过去的几十年里，交换代数不仅在数学的其他研究领域发现了显著的相互作用，而且在其他学科，如工程和计算机科学中证明了一个有价值的工具。为研究生提供建议，指导博士后，组织讲习班和研讨会也是这个项目的一部分。更具体地说，PI将研究局部上同调，de Rham上同调和d模。项目包括发展特征p上的梯度d模的de Rham上同调理论，通过研究与局部环相关的简单复的同调来推广由Faltings和Hartshorne引起的经典连通性定理，将特征p 0上射影方案的Lyubeznik数的研究推广到特征0，将他之前关于局部上同模的有限性性质的研究推广到分支正则环和他之前关于局部上同模的zariski -闭性支持的研究，并研究了特征p上的奇点。本课题所解决的问题将提供特征p上的梯度d模的de Rham上同理论，解决一些长期悬而未决的关于局部上同模的有限性性质的问题。对局部上同模的结构提供了新的认识，并在特征p和特征0的奇点之间建立了新的联系。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。

The proposed project is in the area of commutative algebra, which is a study of systems of algebraic (polynomial) equations with solutions in commutative rings. A commutative ring may be viewed as an abstract analogue of the integers, in which one can add, subtract and multiply elements. In the past few decades, commutative algebra has not only found remarkable interactions with other research areas in mathematics, but proven a valuable tool in other disciplines, such as engineering and computer science. Advising graduate students, mentoring postdocs, and organizing workshops and seminars are also part of this project.More specifically, the PI will study local cohomology, de Rham cohomology and D-modules. Projects include developing a theory of de Rham cohomology of graded D-modules in characteristic p, generalizing classic connectedness theorems due to Faltings and Hartshorne by studying homology of a simplicial complex associated a local ring, extending work on Lyubeznik numbers of projective schemes in characteristic p 0 to characteristic zero, extending his previous work on finiteness properties of local cohomology modules to ramified regular rings and his previous work on Zariski-closedness of support of local cohomology modules, and investigating singularities in characteristic p. Solutions to the problems addressed in this project will provide a theory of de Rham cohomology of graded D-modules in characteristic p, settle some long-standing open questions regarding finiteness properties of local cohomology modules, provide new insight into the structure of local cohomology modules, and lead to new connections between singularities in characteristic p and those in characteristic zero.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.