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，我们可以使用几何学(通过找出直线、平面、曲线的交点)来求解方程。这些想法已经演变成代数几何，这是一个应用范围从物理到网络安全的现代数学领域。在这个合作项目中，我们对与物理的联系特别感兴趣。这些联系是通过“派生类别”建立的，这是一个与几何形状相关的复杂的数学数据系统。虽然派生范畴在近几十年经历了爆炸性的发展，但有关它们的许多问题仍然没有得到解决。这个项目试图使用主要研究人员和其他许多人在过去十年中开发的技术来解决关于派生类别的一些核心和最新的问题。该奖项还将支持本科生和研究生。本项目关注关于派生类别的三个具体问题。总而言之，这些问题包含了许多数学领域，包括二元几何、计数几何和非交换代数。具体地说，我们的第一个问题研究了派生范畴和二元几何之间的联系(以及可能隐藏的翻转/翻转和派生部分紧凑化之间的联系)。我们问K-等价簇是否有等价的派生范畴，这是Bondal-Orlov和Kawamata关于派生范畴的一个中心猜想。我们的第二个问题是问量子上同调的分解如何与派生范畴的半正交分解相关。正如康采维奇和库兹涅佐夫所提出的，这种联系推测是通过镜像对称来实现的。最后，我们的第三个问题是问派生范畴如何解释奇点的分解(即来自非交换代数的模空间)。在这里，我们的目的是利用同调镜像对称的思想来构造非对易的可逆分解。这些决议的存在是范登伯格猜测的。作为一个整体，这个项目的目的是使用几何不变理论、非交换代数、派生代数几何和镜像对称的技术在这三个问题/猜想之间进行内插。中心主题是明确使用和构建傅里叶-穆凯核，其几何为理解这些问题提供了一个立足点。结合主要研究人员过去在傅立叶-Mukai核的构造、派生范畴的墙交叉和虚拟基本循环方面的工作，我们期望增进我们对这些基本问题的理解。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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 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.