Local Cohomology and Related Questions

这个项目是在交换代数和代数几何，领域的解决方案集的多项式方程进行了研究。 这些领域在数学的其他领域和其他学科中有许多应用。 该项目涉及局部上同调的问题，这是交换代数中使用的一个强大工具，并与其他几个数学领域联系起来。 发现两个不同的数学领域之间的联系，有可能通过提供一套新的技术来解决老问题，从而丰富这两个领域。 为学生提供咨询，指导博士后，邀请演讲，并组织一个关于交换代数中D模块的会议也将是这个项目的一部分。在过去的20年里，研究者对局部上同调的研究中，有很大一部分都致力于研究与几个不同数学领域的一些惊人联系。例如，局部上同调提供了一种证明代数簇的拓扑上的不可达结果的方法，而D-模提供了一种证明局部上同调模的有限性的方法。虽然在这一思路方面取得了相当大的进展，但仍存在许多悬而未决的问题。该项目旨在更好地理解一些相互关联的问题，如局部上同调模的结构和算法计算，Lyubeznik数，De Rham同调和代数簇的上同调，直接和项猜想（通过混合特征的局部域的绝对积分闭包）和紧闭包。局部上同调是贯穿所有这些问题并将它们相互联系起来的共同线索。主要的方法是使用D-模和F-模。

This project is in commutative algebra and algebraic geometry, fields where solution sets of polynomial equations are studied. These areas have many applications in other areas of mathematics and in other disciplines. The project is concerned with questions in local cohomology, a powerful tool used in commutative algebra, and connections with several other areas of mathematics. The discovery of a connection between two different areas of mathematics holds a potential for enriching both of them by making available new sets of techniques for attacking old problems. Advising students, mentoring postdocs, giving invited talks, and organizing a conference on D-modules in commutative algebra will also be part of this project. A large part of the investigator's research on local cohomology over the last twenty years has been devoted to the study of a number of striking connections with several quite diverse areas of mathematics. For example, local cohomology provides a way of proving otherwise inaccessible results on the topology of algebraic varieties, while D-modules provide a way of proving otherwise inaccessible finiteness properties of local cohomology modules. While considerable progress on this circle of ideas has been made, many open questions remain. The project is aimed at a better understanding of a number of interrelated problems such as the structure and algorithmic computation of local cohomology modules, Lyubeznik numbers, De Rham homology and cohomology of algebraic varieties, the direct summand conjecture (via the absolute integral closure of a local domain in mixed characteristic), and tight closure. Local cohomology is the common thread that runs through all these problems and connects them to each other. The principal methods to be employed are the use of D-modules and F-modules.