FRG: Collaborative Research: Higher Categorical Structures in Algebraic Geometry

这个项目旨在应用数学的主要新发展来解决代数和代数几何中的开放问题。代数是研究广义系统的数字，而代数几何是关注的几何解决方案的多项式方程。这两个领域都在整个数学中使用，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）中使用的算法定期接触日常生活。该项目还使用了在过去二十年中发展起来的高级范畴理论，这使得系统地处理微妙的、定义松散的对象成为可能。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，一些关于凝聚数学的工作提出了将这种新的控制扩展到密切相关的分析领域的可能性。该项目将利用这一前沿工作，试图解决代数几何中的长期问题，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，并将支持针对早期职业数学家的几个讲习班。首先，PI的目标是找到完整的非交换范畴不变量，并找到一个桥梁，直接从拓扑不变量的范畴。 没有已知的非交换范畴不变量足以重建一个代数簇。在良好的情况下，PI和合作者的工作表明，底层空间足以进行这种重建。接下来，为了澄清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性猜想。第三，PI将研究代数簇的p-adic上同调，通过更高的范畴不变量，如拓扑Hochschild同调，应用于派生范畴。最后，PI将试图表明，最近构建的理论的核模块产生正确的非交换不变量的一个刚性的分析品种，并将旨在推广前三个项目，以更一般的分析context.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.