Homological Aspects of Exterior and Other Power Operations

该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、曲面或更高维度空间上的一个不光滑的地方，也就是说，它有一个尖点(尖点，它折叠或突然转向的地方)或它自己交叉。对奇点的理解在计算机视觉和医学成像中都有应用。一般来说，人们想要找到并研究奇点有多极端的测量方法，以及在变形等操作下这些奇点是如何变化的。这是使用抽象的代数技术来完成的，因为通常情况下，维度、方程的数量、或变量和未知系数使空间完全无法可视化。目标是将不变量附加到奇点，这些奇点是奇点性质的度量。这个项目用三种不同的方式研究这种不变量。更确切地说，这个项目涉及研究外代数、高阶微分模、复数的外幂以及相关的幂运算，如亚当斯运算，有时还包括微分分次代数结构。众所周知，外幂运算的同调行为是复杂的，但这些代数和运算在代数和数学的许多部分都扮演着核心角色。本项目主要集中在以下几个方面的研究方向：(1)Adams运算及其在Hilbert-kunz重数中的应用；(2)p-微分的奇点不变量；(3)分次Artin代数的分辨率。首先，主要目标是开发希尔伯特-昆兹多重性的特征零版本。对于第二部分，重点是通过它们的高阶微分模来研究奇点；目标是通过展示它们的分辨率如何在一般的非孤立Gorenstein奇点环境中相互关联来理解对称性和从它们获得的不变量的消失，例如广义Tjuina数。第三个涉及将Artin代数的解决方案应用于各种问题，如关于Betti数的猜想和在解决方案上寻找dg-结构，以进一步了解Koszul同调。这一奖项反映了NSF的法定使命，并通过使用基金会的智力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.