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 set of 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 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 time is right for a deeper study of these structures. This project will develop such a study through targeted applications to new classes of examples which have been selected for novelty, either in settings or technical details.
The PI will pursue several projects that involve different kinds of categorical and classical symmetries in commutative algebra and algebraic geometry. The general theme of the research is to find and exploit hidden symmetries in a problem to gain new information about it. This information can take the form of a new structural property or a finiteness result about a family of invariants. This research will focus in particular on several classes of mathematical objects for which these properties or calculation of invariants have resisted more traditional means of attack. Among these are: the action of ordered injections on the homology of groups of upper-triangular matrices, the action of surjections on the cohomology of compactified moduli spaces, and Hopf ring actions on syzygies of secant varieties of varieties arising from multilinear algebra. These are three examples of variants of the recently well-studied "FI-modules," which offer new technical challenges and open the door to new paradigms of "representation stability."
