Homological Commutative Algebra and Symmetry

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

The proposed research aims to investigate subtle interactions between commutative algebra and classical algebraic geometry on the one hand, and representation theory and modular arithmetic on the other hand. The main subject of exploration is homology, which can be thought of as the study of (dis)similarities. Despite being regarded as an abstract mathematical tool, homology has found many applications in recent years, particularly to data analysis and computer science. In algebra, homology refers to a way of measuring the difference between an implicitly defined set of objects (cycles) and an explicitly defined subset (of boundaries). In geometry, it is a tool used to distinguish between different shapes. The homology attached to an object usually focuses on the most significant traits, and it reflects its symmetries in intriguing ways. The PI will investigate homological theories associated to flag varieties, which are geometric objects parametrizing increasing sequences of subspaces of linear spaces, such as a point contained in a line contained in a plane. The symmetry comes from moving the linear spaces around while preserving their containment relations. The homological theories considered depend on a prime number p=2,3,5,7,11 etc., and the goal is to understand how the resulting homology depends on p, and how it encodes the symmetries of the flag varieties. This can then be further applied to study algebro-geometric objects of interest, such as matrices and higher-dimensional tensors. The proposed research is suitable for engaging students in research, as well as for computer experimentation and software development. A fundamental question at the confluence of commutative algebra, algebraic geometry and representation theory is to describe the cohomology of line bundles on flag varieties. A well-known case is that of the projective space, where it is equivalent to computing local cohomology for a polynomial ring with support in the ideal of the variables. Other examples include Grassmannians, complete flag varieties, or the incidence correspondence. For flag varieties over a field of characteristic zero the cohomology is computed by the Borel-Weil-Bott theorem, but the positive characteristic problem remains wide open. The PI's goal is a systematic study of this question, with an emphasis on concrete examples, and on showcasing peculiar interactions between algebra, geometry, and symmetry. The PI plans to use the newly acquired knowledge to solve questions of a homological nature regarding fundamental objects such as matrix determinantal varieties and analogues for higher tensors, equations and syzygies, or Koszul modules. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.