Interactions between Commutative Algebra and Representation Theory

University Of California-San Diego, La Jolla CA

Investigators

Abstract

An important theme in mathematics is to find finite descriptions for objects that a priori contain an infinite amount of information. For example, an infinite sequence of numbers might be compactly encoded as the values of a simple function. One such occurrence relevant to this project is the dimensions of a sequence of spaces. In many cases of interest, it may not be possible to directly compute these numbers, but one can instead hope to analyze their rate of growth. An old theme in mathematics has been to understand such problems by finding some algebraic structure on the sequence. Recently, several groups of researchers have found new, exotic algebraic structures that apply to previously unexpected examples of such sequences in areas such as topology, combinatorics, and algebraic geometry. The purpose of this project is to study and uncover the fundamental properties of these new algebraic structures by applying and combining tools from commutative algebra and representation theory. The PI proposes to work on several projects at the interface between commutative algebra and representation theory, specifically the study of new classes of algebraic objects (e.g., twisted commutative algebras, Delta-modules, and FI-modules) and their applications to classical topics (syzygies of algebraic varieties, homological stability, homology of arithmetic groups, etc.). Additionally, the PI will work on problems in Boij-Söderberg theory, which is the study of algebraic invariants, such as Betti tables of graded modules and cohomology tables of vector bundles, "up to scalar multiple." This topic is connected to the general theme in indirect and subtle ways that are still not well understood. The basic theme is to identify a class of examples that have the structure of a module over one of these algebraic structures and to prove that it is finitely generated. The finite generation reveals some information akin to an existence result, but often the results do not come with bounds or any sharper information, so should be seen as a first step in a larger program. The PI proposes to import ideas from commutative algebra and homological algebra, such as Hilbert functions, projective resolutions, Castelnuovo-Mumford regularity, and Koszul duality, to further enhance the understanding of such examples and to reveal new information.