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

University Of Utah, Salt Lake City UT

Investigators

Abstract

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.