Studies in Commutative Algebra

研究者将研究交换环理论中由紧闭理论、交多重理论、局部上同调模的研究以及局部环的同调猜想引起的一些问题。在紧闭理论中，对f正则环的理解和对这类环的定理的发展有着实质性的关注。局部上同模的研究解决了这些模在混合特征正则环上的有限性问题，以及另一个与实闭理论有关的问题。Hilbert-Kunz复数和Tor函子刚性的研究将继续与Claudia Miller合作进行。交换代数是一个与代数几何密切相关的领域：代数几何关注多项式方程解集的几何，而交换代数的主要研究对象是这些解集上的某些函数。代数和几何之间的联系很大程度上要归功于格罗滕迪克、塞尔和扎里斯基的革命性工作。交换代数中一个活跃的研究领域是由Hochster和Huneke发展的紧闭理论。这个理论为现有的定理提供了更有力的表述，并汇集了许多看似无关的问题。它与几何物体奇点的研究也有着密切的联系。局部上同调理论是由格罗滕迪克提出的，他利用它得到了几个惊人的结果。局部上同调可以应用于基本而深刻的问题，例如确定定义代数集所需的多项式方程的最小数量。它继续发展与其他几个数学分支的迷人的相互作用。

The investigator will work on some questions in the theory of commutative rings that arise from tight closure theory, the theory of intersection multiplicities, a study of local cohomology modules, and from the homological conjectures for local rings. In tight closure theory, there is a substantial focus on understanding the class of F-regular rings and on developing theorems for this class of rings. The study of local cohomology modules addresses finiteness issues of these modules over regular rings of mixed characteristic, and another question that is related to the theory of solid closure. Research on Hilbert-Kunz multiplicities and on the rigidity of the Tor functor will be carried out in continued collaboration with Claudia Miller. Commutative algebra is a field closely related to algebraic geometry: while algebraic geometry focuses on the geometry of solutions sets of polynomial equations, in commutative algebra the main objects of study are certain functions on these solution sets. The connections between algebra and geometry are largely due to the revolutionary work of Grothendieck, Serre and Zariski. An active area of research in commutative algebra today is the theory of tight closure developed by Hochster and Huneke. This theory provides stronger formulations of existing theorems, and brings together many seemingly unrelated problems. It also has strong connections with the study of singularities of geometric objects. The theory of local cohomology was developed by Grothendieck who used it to obtain several striking results. Local cohomology has applications to basic, and yet deep, questions such as determining the minimal number of polynomial equations needed to define an algebraic set. It continues to develop a fascinating interaction with several other branches of mathematics.