Collaborative Research: Derived Categories in Birational Geometry, Enumerative Geometry, and Non-commutative Algebra

合作研究：双有理几何、枚举几何和非交换代数中的派生范畴

• 批准号: 2302262
• 负责人: David Favero
• 金额: $ 28万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302262&HistoricalAwards=false
• 关键词: Collaborative; Research; Derived; Categories; Birational

Solutions to polynomial equations form geometric shapes (for example lines, planes, curves). Thus, we can solve equations using geometry (by finding where lines, planes, curves intersect). These ideas have evolved into algebraic geometry, a field of modern mathematics with applications ranging from physics to cybersecurity. In this collaborative project we are particularly interested in the connections to physics. These connections are made through “derived categories”, a complex system of mathematical data associated to a geometric shape. While derived categories have experienced explosive development in recent decades, many questions about them remain unsolved. This project attempts to solve some of the central and most recent questions about derived categories using techniques developed by the principal investigators and many others over the last decade. This award will also support undergraduate and graduate students. This project focuses on three specific questions about derived categories. Together, these questions incorporate numerous areas of mathematics including birational geometry, enumerative geometry, and non-commutative algebra. Specifically, our first question studies connections between derived categories and birational geometry (and the possibly concealed connection between flips/flops and derived partial compactifications). We ask whether K-equivalent varieties have equivalent derived categories, a central conjecture about derived categories due to Bondal-Orlov and Kawamata. Our second question asks how decompositions of quantum cohomology are related to semi-orthogonal decompositions of derived categories. This connection is conjecturally made through mirror symmetry as proposed by Kontsevich and Kuznetsov. Finally, our third question asks how derived categories shed light on resolutions of singularities (namely as moduli spaces coming from non-commutative algebra). Here, we aim to construct non-commutative crepant resolutions using ideas from homological mirror symmetry. The existence of these resolutions has been conjectured by Van den Bergh. As a whole, this project aims to interpolate between these three questions/conjectures using techniques from geometric invariant theory, non-commutative algebra, derived algebraic geometry, and mirror symmetry. The central theme is the explicit use and construction of Fourier-Mukai kernels whose geometry provides a foothold into understanding these problems. Combining the past work of the principal investigators on the construction of Fourier-Mukai kernels, wall crossing for derived categories, and virtual fundamental cycles, we expect to advance our understanding of these fundamental questions.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.

多项式方程的解决方案形成几何形状（例如线，平面，曲线）。那就可以使用几何形状（通过查找线，平面，曲线相交的位置）求解方程。这些想法已经演变成代数几何，这是一个现代数学领域，其应用从物理学到网络安全。在这个协作项目中，我们对与物理的联系特别感兴趣。这些连接是通过“派生类别”建立的，这是一个与几何形状相关的数学数据系统的复杂系统。尽管衍生的类别在最近几十年中经历了爆炸性的发展，但有关它们的许多问题仍未解决。该项目试图使用主要研究人员和许多其他人在过去十年中开发的技术解决有关派生类别的一些中心和最新问题。该奖项还将支持本科生和研究生。该项目着重于有关派生类别的三个特定问题。总之，这些问题结合了许多数学领域，包括生物几何形状，枚举几何学和非共同代数。具体而言，我们的第一个问题研究是衍生类别与生物几何形状之间的联系（以及fllips/flops和派生的部分压缩之间可能的隐蔽连接）。我们询问K-等效品种是否具有等效的衍生类别，这是一个核心类别的核心概念，该类别是由于邦德 - 奥洛夫和卡瓦玛塔而引起的。我们的第二个问题询问量子共同体的分解与派生类别的半正交分解有关。这种连接是通过Kontsevich和Kuznetsov提出的镜像对称性进行猜想的。最后，我们的第三个问题询问派生类别是如何阐明奇异性分辨率的（即来自非共同代数的模量空间）。在这里，我们旨在使用同源镜子对称性中的思想来构建非交流性的毛茸茸的决议。这些决议的存在是由van den Bergh猜想的。总体而言，该项目旨在使用几何不变理论，非交通式代数，派生的代数几何形状和镜像对称性的技术插值这三个问题/猜想。中心主题是明确使用和构建傅里叶 - 木核，其几何形状为理解这些问题提供了立足点。结合了主要研究人员在建造傅立叶核内核，衍生类别的墙壁交叉以及虚拟基本周期的过去工作，我们希望提高我们对这些基本问题的理解。该奖项反映了NSF的法定任务，并通过该基金会的知识分子优点和广泛的影响来评估NSF的法定任务，并被认为是值得的支持。