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

多项式方程的解形成几何形状（例如直线、平面、曲线）。因此，我们可以用几何学来解方程（通过找到直线、平面、曲线的交点）。这些想法已经演变成代数几何，这是现代数学的一个领域，其应用范围从物理学到网络安全。在这个合作项目中，我们对与物理学的联系特别感兴趣。这些联系是通过“衍生类别”建立起来的，这是一个与几何形状相关的复杂数学数据系统。虽然衍生类别在近几十年经历了爆炸性的发展，但关于它们的许多问题仍未得到解决。这个项目试图解决一些核心的和最新的问题，有关衍生类别使用的技术发展的主要研究人员和许多其他人在过去的十年。该奖项也将支持本科生和研究生。这个项目主要关注关于衍生类别的三个具体问题。这些问题结合了许多数学领域，包括两国几何、枚举几何和非交换代数。具体来说，我们的第一个问题研究了派生范畴和双几何之间的联系（以及flip /flops和派生的部分紧化之间可能隐藏的联系）。我们问k -等价变体是否有等价的派生范畴，这是由Bondal-Orlov和Kawamata提出的一个关于派生范畴的中心猜想。我们的第二个问题是量子上同调的分解是如何与派生范畴的半正交分解联系起来的。据推测，这种联系是通过Kontsevich和Kuznetsov提出的镜像对称来实现的。最后，我们的第三个问题是，衍生范畴如何阐明奇点的解析（即来自非交换代数的模空间）。在这里，我们的目标是利用同调镜像对称的思想构造非交换的蠕变分辨率。这些决议的存在是由范登伯格推测出来的。作为一个整体，该项目旨在利用几何不变理论、非交换代数、衍生代数几何和镜像对称的技术，在这三个问题/猜想之间进行插值。中心主题是傅里叶-穆凯核的明确使用和构造，其几何为理解这些问题提供了立足点。结合过去主要研究人员在傅里叶-穆凯核的构造、派生类别的壁交叉和虚拟基本循环方面的工作，我们期望推进我们对这些基本问题的理解。该项目由代数和数论项目和促进竞争研究的既定项目（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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 project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).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.