Homological Aspects of Exterior and Other Power Operations

这个奖项支持对代数几何的一个基本问题的研究：如何找到奇点的度量？奇点是曲线、曲面或高维空间上的一个不光滑的地方，也就是说，它有一个尖（一个尖锐的点，它折叠或突然转向的地方）或它自己交叉。对奇点的理解可以应用于计算机视觉和医学成像。一般来说，人们想要找到并研究奇点的极端程度，以及奇点在变形等操作下如何变化。这是使用抽象的代数技术完成的，因为通常维度，或方程的数量，或变量和未知系数使空间完全无法可视化。我们的目标是将不变量附加到奇点上，作为奇点本质的度量。本项目以三种不同的方式研究这些不变量。更准确地说，这个项目涉及到外部代数、高阶微分的模、复合体的外部幂和相关的幂运算，如Adams运算，有时也涉及到微分梯度代数结构。众所周知，外部幂运算的同调行为是复杂的，然而这些代数和运算在代数和数学的许多部分中起着核心作用。本项目主要研究方向为：(1)Adams运算及其在Hilbert-Kunz多重中的应用；(2)p-微分的奇异不变量；(3)分级Artinian代数的解析。对于第一个，主要目标是开发Hilbert-Kunz多重性的特征零版本。第二，重点是通过奇点的高微分模来研究奇点；目标是通过展示它们的分辨率在一般非孤立的Gorenstein奇点设置下是如何相互关联的，来理解对称性和从中得到的不变量的消失，例如广义Tjurina数。第三部分涉及将阿提尼代数的解析应用于各种问题，例如关于贝蒂数的猜想和在解析上寻找g结构，以便进一步理解科祖尔同调。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.