Homological Aspects of Exterior and Other Power Operations

该奖项支持研究代数几何的基本问题之一：如何找到奇点的措施？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，它有一个尖点（一个尖锐的点，一个突然折叠或转弯的地方）或它自己交叉。对奇点的理解在计算机视觉和医学成像方面有应用。一般来说，人们希望找到并研究奇点的极端程度以及这些在变形等操作下如何变化的措施。这是使用抽象代数技术来完成的，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不切实际地可视化。我们的目标是将不变量附加到奇点上，这些奇点是奇点性质的度量。这个项目以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，高微分模，复合物的外幂，以及相关的幂运算，如亚当斯运算，有时微分分次代数结构发挥作用。已知外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。本计画的研究方向为：（1）亚当斯运算及其在希尔伯特-昆兹多重性上的应用;（2）p-微分奇异性的不变量;（3）分次阿廷代数的分解。对于第一个，主要目标是开发一个特征零版本的希尔伯特-昆兹多重性。对于第二个，重点是研究奇点通过其模块的高微分;目标是了解对称性和消失的不变量从他们获得的，如广义Tjurina数，通过显示他们的决议是如何相互关联的一般非孤立Gorenstein奇点设置。第三个奖项涉及将Artinian代数的分解应用于各种问题，例如Betti数的解析和在分解上寻找dg结构，以进一步理解Koszul同源性。该奖项反映了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.