FRG: Collaborative Research: Higher Categorical Structures in Algebraic Geometry

FRG：合作研究：代数几何中的更高范畴结构

• 批准号: 2152088
• 负责人: Andrei Caldararu
• 金额: $ 34.14万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2152088&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Higher; Categorical

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.

该项目旨在将数学方面的主要新发展应用于代数和代数几何形状的问题。代数是对数量的广义系统的研究，而代数几何形状与多项式方程溶液的几何形状有关。这两个字段均在整个数学过程中使用，并通过计算机视觉中使用的算法定期触摸日常生活（例如，在手机摄像机中），卫星通信（错误校正代码）和安全消息传递（使用椭圆形曲线的密码学）。该项目还使用了过去二十年来开发的较高类别理论，这使得有可能系统地处理微妙的，松散定义的对象。这种额外的灵活性导致对代数几何形状中使用的基本对象的新控制。最近，一些关于凝结数学的工作增加了将这种新控制扩展到紧密相关的分析领域的可能性。该项目将利用这项尖端的工作来试图解决代数几何形状中的长期问题，并在分析代数几何形状中介绍和解决新问题。它将为研究生和博士后研究人员提供研究和培训机会，并将支持针对早期数学家的几个研讨会。该项目提出了四个主要的研究挑战。首先，PI的目的是找到完整的非交通性分类不变式，并直接从拓扑不变的桥上找到一座桥。 没有已知的非交通性分类不变性足以重建代数品种。在好的情况下，PIS和合作者的工作表明，基础空间足以进行这种重建。接下来，为了阐明不交流对象内的交换对象的作用，PI将研究DG类别的变形和局部系统，以解决Orlov的几何形状猜想。第三，PI将通过更高的分类不变式（例如拓扑Hochschild同源性）研究代数品种的P-ADIC共同体，应用于派生类别。最后，PI将试图表明，最近建造的核模块理论产生了刚性分析多样性的正确非交换性不变性，并将旨在将前三个项目推广到更一般的分析环境中。该奖项反映了NSF的法定任务，并通过使用基金会的知识优点和广泛的影响来评估通过评估来进行评估，并通过评估值得进行评估。