FRG: Collaborative Research: Higher Categorical Structures in Algebraic Geometry

该项目旨在应用数学的重大新进展来解决代数和代数几何中的问题。代数是对广义数字系统的研究，而代数几何则涉及多项式方程解的几何形状。这两个领域都在数学中使用，并通过计算机视觉（例如手机摄像头）、卫星通信（纠错码）和安全消息传递（使用椭圆曲线的密码学）中使用的算法定期触及日常生活。该项目还使用了过去二十年发展起来的更高范畴理论，这使得系统地处理微妙的、松散定义的对象成为可能。这种额外的灵活性导致对代数几何中使用的基本对象的新控制。甚至最近，一些关于浓缩数学的工作提出了将这种新控制扩展到密切相关的分析领域的可能性。该项目将利用这项前沿工作尝试解决代数几何中长期存在的问题，并引入和解决解析代数几何中的新问题。它将为研究生和博士后研究人员提供研究和培训机会，并将支持多个针对早期职业数学家的研讨会。该项目解决了四个主要研究挑战。首先，PI 的目标是找到完全的非交换范畴不变量，并找到直接从拓扑不变量到范畴不变量的桥梁。 没有已知的非交换分类不变量足以重建代数簇。在良好的情况下，PI 和合作者的工作表明底层空间足以进行此类重建。接下来，为了阐明非交换对象中交换对象的作用，PI将研究dg范畴的变形和局部系统，试图解决奥尔洛夫的几何猜想。第三，PI 将通过应用于派生类别的拓扑 Hochschild 同调等更高的类别不变量来研究代数簇的 p 进上同调。最后，PI 将试图证明最近构建的核模块理论产生了严格分析类型的正确非交换不变量，并将旨在将前三个项目推广到更一般的分析背景。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 project aims to apply major new developments in mathematics to open questions in algebra and algebraic geometry. Algebra is the study of generalized systems of numbers, while algebraic geometry is concerned with the geometry of solutions of polynomial equations. Both fields are used throughout mathematics and touch regularly on daily life via algorithms used in computer vision (for instance in cell phone cameras), satellite communications (error-correcting codes), and secure messaging (cryptography using elliptic curves). The project also uses higher category theory developed over the last two decades, which makes it possible to systematically deal with subtle, loosely defined objects. This extra flexibility leads to new control over the basic objects used in algebraic geometry. Even more recently, some work on condensed mathematics raises the possibility of extending this new control to closely related areas of analysis. This project will use this cutting-edge work to attempt to settle longstanding questions in algebraic geometry and to introduce and solve new questions in analytical algebraic geometry. It will provide research and training opportunities for graduate students and postdoctoral researchers and will support several workshops aimed at early-career mathematicians.There are four main research challenges addressed by this project. First, the PIs aim to find complete noncommutative categorical invariants and to find a bridge directly from the topological invariants to the categorical ones. No known noncommutative categorical invariant suffices to reconstruct an algebraic variety. In good cases, work of the PIs and collaborators shows that the underlying space is enough for such a reconstruction. Next, to clarify the role of commutative objects inside noncommutative objects, the PIs will study the deformations and local systems of dg categories in an attempt to settle Orlov's geometricity conjecture. Third, the PIs will study the p-adic cohomology of algebraic varieties via higher categorical invariants such as topological Hochschild homology, applied to the derived category. Finally, the PIs will try to show that the recently constructed theory of nuclear modules yields the correct noncommutative invariants of a rigid analytic variety and will aim to generalize the first three projects to the more general analytic context.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.