Homological Aspects of Exterior and Other Power Operations

Syracuse University, Syracuse NY

Investigators

Abstract

This award supports research into one of the fundamental questions of algebraic geometry: How does one find measures of a singularity? A singularity is a place on a curve, surface, or higher dimensional space where it is not smooth, that is, it has a cusp (a sharp point, a place where it folds or turns abruptly) or it crosses itself. The understanding of singularities has applications to computer vision and medical imaging. In general, one wants to find and study measures of how extreme a singularity is and how these vary under operations such as deformations. This is done using abstract algebraic techniques as usually the dimension, or the number of equations, or variable and unknown coefficients make the spaces completely impractical to visualize. The goal is to attach invariants to singularities that are measures of the nature of the singularity. This project studies such invariants in three different ways. More precisely, this project involves the study of the exterior algebra, modules of higher differentials, exterior powers of complexes, and related power operations such as Adams operations, sometimes with differential graded algebra structures playing a role. The homological behavior of exterior power operations is known to be complicated, yet these algebras and operations play a central role throughout many parts of algebra and mathematics. This project focuses on the following research directions (1) Adams operations and applications to Hilbert-Kunz multiplicities, (2) invariants of singularities for p-differentials, and (3) resolutions of graded Artinian algebras. For the first, the main goal is to develop a characteristic zero version of Hilbert-Kunz multiplicity. For the second, the focus is on studying singularities via their modules of higher differentials; the goal is to understand symmetries and the vanishing of invariants obtained from them, such as generalized Tjurina numbers, by showing how their resolutions are interrelated in the general non-isolated Gorenstein singularity setting. The third involves applying the resolution of Artinian algebras to various problems, such as conjectures on Betti numbers and finding dg-structures on resolutions in order to further understand Koszul homology. 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.