Employing Symmetry in Commutative Algebra

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

The project pertains primarily to the field of commutative algebra and is concerned with the study of systems of polynomial equations: these equations should be thought of as the algebraic relationships between the set of parameters in some model. Oftentimes mathematical models of natural phenomena come equipped with a large group of symmetries, and the main goal of the current project is to develop techniques that employ the symmetry in order to reveal both qualitative and quantitative features of the models. The questions and the methods are strongly influenced by classical algebraic geometry, which analyzes the models from a geometric perspective, by representation theory, which is an algebraic study of symmetry, and by D-module theory, which is an algebraic perspective on the infinitesimal study of the models. Computer experimentation is important for the success of the project. The investigator, together with students and collaborators, will develop and disseminate the necessary computer algebra software and will make available the results of experimentation that are of general interest. This research project investigates a number of open questions concerning fundamental invariants associated to algebraic varieties endowed with a large group of symmetries, by analyzing the implications of the symmetries to the structure of the invariants. The main emphasis will be on the study of syzygies and local cohomology groups, with a view towards understanding further invariants such as regularity and projective dimension, cohomological dimension and arithmetic rank, or Lyubeznik numbers and Bernstein-Sato polynomials. The project will combine methods from commutative algebra and classical algebraic geometry, as well as techniques from representation theory and the theory of D-modules. For instance, the investigator will explore the D-module structure of the local cohomology groups associated to natural stratifications of the spaces of binary forms. On a separate note, the investigator will research the connections between syzygies of ideals in rings of polynomial functions on spaces of matrices, and representations of general linear Lie superalgebras.