Homological approaches to differential forms, differential operators, and transfer of algebra structures

代数几何的主要问题之一是：如何找到奇异性的措施？奇点是曲线、曲面或高维空间上不光滑的地方，即，它有一个尖锐的点或交叉本身。一般来说，人们希望能够找到奇点有多极端以及这些奇点在地图下如何变化的度量，但通常方程或变量的数量使得图形不可能或误导。这是解决采用结构骨干给出的代数几何和交换代数。一般来说，目标是将不变量附加到奇点上，这些不变量是奇点特征的度量。奇异性将研究在几个方面：通过微分形式和微分算子，其特点是光滑;并通过构建新的代数结构，不仅产生不变量，但也工具，以发现新的不变量。从长远来看，对奇点的研究可以应用于计算机视觉和医学成像，以及物理学中的弦理论。该项目将继续PI在研究生教育与博士生的强烈参与。PI将继续共同组织会议和研讨会，如2024年MSRI学期课程的介绍性研讨会和陈省身研究所未来的MSRI夏季研究生院。交换代数和同调代数是发展代数几何基础的关键。这个项目将解决一些中心主题和它们的相互作用：微分形式，微分算子，余切复形，dg-代数和A-无穷结构。重点是（1）微分形式和余切复形的分解，（2）固定阶微分算子及其分解，（3）分次Artin代数的DG代数分解，以及（4）通过A-无穷大结构比较棒和Eagon分解。对于第一，重点是了解对称性和消失的不变量从更高的微分，如广义Tjurina数，通过显示它们的决议是如何相互关联的Gorenstein奇点。对于第二个，一个新的同调方法将被用来寻找超曲面的微分算子。第三个涉及使用HPT（同调扰动理论）转移代数结构，和第四个使用A-无穷大结构的关系和推广两个经典的决议。统一的主题是使用同调的方法来深入了解这些问题。核心的挑战是理解外部和对称幂运算的同调行为，这是已知的更多涉及和更少理解，尽管外代数和幂运算在代数和几何的许多部分中起着核心作用，但该奖项反映了NSF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进行评估，被认为值得支持的搜索.

One of the main questions of algebraic geometry is: How does one find measures of singularity? A singularity is a place on a curve, surface, or higher dimensional space where it is not smooth, i.e., it has a sharp point or crosses itself. In general, one wants to be able to find measures of how extreme a singularity is and how these vary under maps, but usually the number of equations or variables makes a graph impossible or misleading. This is solved by employing the structural backbone given by algebraic geometry and commutative algebra. Generally, the goal is to attach invariants to singularities that are measures of the character of the singularity. Singularities will be studied in several ways: via differential forms and differential operators, which characterize smoothness; and by constructing new algebraic structures which yield not only invariants but also tools to discover new invariants. In the long run, work on singularities can lead to applications to computer vision and medical imaging and to string theory in physics. The project will continue the PI’s strong engagement in graduate education with PhD students. The PI will continue to co-organize conferences and workshops such as the Introductory Workshop at an MSRI semester program in 2024 and a future MSRI Summer Graduate School at the Chern Institute. Commutative and homological algebra are crucial in developing the foundations of algebraic geometry. This project will address some central topics and their interplay: differential forms, differential operators, cotangent complexes, and dg-algebra and A-infinity structures. The focus will be on (1) resolutions of differential forms and the cotangent complex, (2) differential operators of fixed orders and their resolutions, (3) DG algebra resolutions of graded Artinian algebras, and (4) comparisons of bar and Eagon resolutions via A-infinity structures. For the first, the focus is on understanding symmetries in and vanishing of invariants obtained from higher differentials, such as generalized Tjurina numbers, by showing how their resolutions are interrelated for Gorenstein singularities. For the second, a new homological approach will be used to work on finding the differential operators of hypersurfaces. The third involves the use of HPT (homological perturbation theory) to transfer algebra structures, and the fourth use of A-infinity structures to relate and generalize two classical resolutions. The unifying theme is to use homological methods to gain insight into these problems. The central challenge is to understand the homological behavior of exterior and symmetric power operations, which is known to be more involved and less understood, despite the fact that exterior algebras and power operations play a central role throughout many parts of algebra and geometry.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.