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

建议的项目是在交换代数领域，它是研究在交换环中有解的代数(多项式)方程组。交换环可以看作是整数的抽象类比，其中可以对元素进行加、减、乘运算。在过去的几十年里，交换代数不仅与数学中的其他研究领域产生了显著的相互作用，而且在其他学科中被证明是一个有价值的工具，如工程和计算机科学。指导研究生，指导博士后，组织研讨会和研讨会也是这个项目的一部分。更具体地说，PI将学习局部上同调，de Rham上同调和D-模。这些工作包括：发展了特征p中分次D-模的de Rham上同调理论；通过研究局部环上的单纯形复形的同调，推广了Faltings和HartShorne的经典连通性定理；将特征p上投射方案的Lyubeznik数的工作推广到特征零；将他以前关于局部上同调模的有限性质的工作推广到了分枝正则环上，并研究了局部上同调模的支撑性的Zariski-闭性；研究了特征p中的奇点问题.这些问题的解决将为特征p中分次D-模的de Rham上同调提供一个理论，解决一些长期悬而未决的关于局部上同调模的有限性质的问题，为局部上同调模块的结构提供了新的见解，并导致了特征p中的奇点和特征零中的奇点之间的新联系。该奖项反映了NSF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。

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.