Homological Aspects of Commutative Algebra and Applications to Modular Representation Theory

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One of the myriad functions of mathematics is to provide language to formulate, and tools to solve, equations that describe the physical world. Often these equations are algebraic in nature, such as those describing lines, circles, and parabolas. Typically, the equations have infinitely many solutions -- think about the equation defining a circle -- and it is usually not possible to write down a complete list of solutions. Rather, the objective is to find ways to study the structure of the collection of the solution set, which is called a variety. A fruitful approach has been to consider the (algebraic) functions on the variety. One part of this research project addresses questions that have emerged in this endeavor. Mathematics has also been remarkably successful in describing and studying phenomenon related to symmetry. This leads to a mathematical structure called a group. Intriguingly, in certain contexts, there is a way to attach a variety to a group, and in the past few years various researchers, including the investigator, have been able to solve problems related to groups using tools that had been developed to study varieties. A part of the current project deals with these aspects.This research is rooted in commutative algebra, with applications also to the representation theory of finite group schemes in positive characteristic. The first part of the project concerns the homological aspects of modules over commutative rings. The problems posed range from those that have arisen from the internal developments in the subject to ones inspired by recent advances in the broader mathematical context, notably the representation theory of algebras and homotopy theory. The focus in the second part of the project is on the Hochschild cohomology of commutative algebras. Classically, Hochschild cohomology has been of interest for it is one of two main cohomology theories that capture properties of the diagonal morphism. A different aspect of Hochschild cohomology has begun to play an increasingly important role: Its action on derived categories, and various triangulated categories. The problems proposed in the second part of the project address both aspects. In recent years, notions and techniques from commutative ring theory have proved to be remarkably effective in solving problems in, and shedding new light on, the representation theory of finite groups over a field of positive characteristic. This project will develop these connections in new directions.