Local Cohomology, the Frobenius Endomorphism, D-Module Theory, and Invariant Theory

众所周知，日常生活中遇到的许多几何物体都可以用方程来描述。 此外，基本和有趣的对象通常可以使用多项式方程来描述（例如，考虑熟悉的圆方程）。 虽然多项式是基于一些最基本的运算（即加法和乘法），但它们能够描述各种各样的现象，有时可以以神秘而复杂的方式表现。 本研究计画是关于交换代数的数学领域，也就是多项式方程式的研究。 这个领域有许多重要的应用;例如，它用于密码学和物理学。 该项目的主要目标是了解在多项式方程研究中出现的数学结构。 在研究项目中采用的许多方法涉及其他数学领域的技术。 这个项目的目标是推进交换代数中一些基本对象的理解。 特别是，它的目的是阐明结构的局部上同调模，环的混合特征，伯恩斯坦-佐藤多项式，和奇异性的特征p.调查特别是动机这些主题之间的深刻联系。 本项目的主要目的是：（1）利用群作用和D-模理论研究局部上同调模;（2）利用Lyubeznik数（以及研究者和合作者最近介绍的变体）比较相等特征和混合特征的环，并理解混合特征的环;以及（3）找到F-阈值的显式公式并使用这些公式来产生Bernstein-Sato多项式的根。 所采用的技术使用不变理论，非交换代数和组合。

It is well known that many of the geometric objects encountered in daily life can be described using equations. Furthermore, fundamental and interesting objects can often be described using polynomial equations (for example, consider the familiar equation for a circle). Though polynomials are based upon some of the most elementary operations (namely, addition and multiplication), they are able to describe a rich variety of phenomena, and can sometimes behave in mysterious and complicated ways. This research project concerns the mathematical field of commutative algebra, that is, the study of polynomial equations. This field has many important applications; for example, it is used in cryptography and physics. The broad goal of this project is to understand mathematical structures that arise in the study of polynomial equations. Many of the methods employed in the research project involve techniques from other areas of mathematics. The goal of this project is to advance the understanding of some fundamental objects in commutative algebra. In particular, it aims to shed light on the structure of local cohomology modules, rings of mixed characteristic, Bernstein-Sato polynomials, and singularities in characteristic p. The investigator is especially motivated by the deep connections between these topics. The project seeks to do the following: (1) to study local cohomology modules using group actions and D-module theory; (2) to use the Lyubeznik numbers (and variants introduced recently by the investigator and collaborator) to compare rings of equal characteristic and mixed characteristic, and to understand rings of mixed characteristic; and (3) to find explicit formulas for F-thresholds and use these to produce roots of the Bernstein-Sato polynomial. The techniques employed use invariant theory, noncommutative algebra, and combinatorics.