Homological Algebra of Landau-Ginzburg Mirror Symmetry

这是一个研究项目，旨在建立代数和几何结果，其灵感来自弦理论的对偶性。就在世纪之交之前，理论物理学为几何学提供了一种洞察力，导致了许多现代几何研究的成功。他们发现弦理论中的对偶性对高维几何的研究有着重要的意义和应用，使得关于某些六维空间几何的经典问题的答案变得容易获得。粗略地说，（经典）弦理论要为宇宙提供一个潜在的物理理论，它需要宇宙是10维的。其中4个维度是我们在生活中经历的标准3个空间维度和1个时间维度，其他6个维度是所谓的卡-丘流形。目前还不清楚到底有多少卡-丘流形，我们也用许多不同的方法来研究它们，但弦理论让我们在几何研究学科之间建立了深刻的联系。特别是，弦论中的对偶性指出，每个卡-丘流形都有一个“镜像”，它是另一个卡-丘流形，因此一个流形的各种几何和物理性质被封装在它的镜像的其他几何和物理性质中。这种现象在数学中被称为镜像对称。特别是，与卡-丘流形相关的辛几何中的困难计算和计算开放问题现在被编码在其镜像的代数几何中。在镜像对称开始时，这些代数几何计算要容易得多，然后它们被用作我们在辛几何中要证明的东西的指导原则。这使得枚举几何中的百年难题得以实现。1994年，菲尔兹奖获得者孔采维奇提出了镜像对称的一个理论上但完全数学化的版本，将辛几何编码在所谓的福谷范畴中，将代数几何编码在相干层的导出范畴中。在过去的三十年里，镜像对称已经扩展，现在可以看到镜像对称不仅仅是卡-丘流形之间的关系，而是更多的几何空间（例如，Fano流形，log Calabi-Yau簇）。然而，它也被扩展到奇点的研究。有趣的是，我们可以通过构造一个函数来模拟某些空间的几何形状，使得该函数沿原始空间沿着是奇异的。然后我们可以变形这个模型，仍然得到弦理论的物理模型。这是朗道-金兹伯格模型的一个例子。朗道-金兹伯格模型的镜像对称性已经在少数情况下建立，并且在研究经典的高维形状（如卡-丘流形）时表现出强大的功能。然而，在朗道-金兹伯格模型的镜像对称性研究中，仍然有一些基础问题需要解决。理想情况下，我们想证明Landau-Ginzburg模型的Kontsevich猜想的一种形式，但在我们这样做之前，我们需要理解Landau-Ginzburg模型的代数几何方面。这个项目旨在更好地理解Landau-Ginzburg模型的这种分类观点，证明上面的相干层的衍生类别（称为（矩阵）因子分解类别）的类似物的各种结构结果。

This is a research project to establish algebraic and geometric results inspired by a duality originally from string theory.Right before the turn of the century, theoretical physics provided an insight to geometry that led to many modern successes in geometric research. They discovered a duality in string theory had implications and applications to the study of higher dimensional geometry, making answers to classical questions about the geometry of certain six-dimensional spaces accessible. Roughly speaking, for (classical) string theory to provide a potential physical theory for the universe, it requires the universe to be 10-dimensional. Four of these dimensions are the standard 3 space dimensions and one time dimension we experience in our lives, and the other six are a so-called Calabi-Yau manifold. It is still unclear how many Calabi-Yau manifolds there are, and we study them in many different ways, but string theory has given us a deep connection between geometric research disciplines. In particular, a duality in string theory states that each Calabi-Yau manifold has a "mirror" which is another Calabi-Yau manifold so that various geometric and physical properties of one are encapsulated in other geometric and physical properties of its mirror. This phenomenon in mathematics is now known as mirror symmetry. In particular, hard computations and computational open questions in symplectic geometry associated to a Calabi-Yau manifold were now encoded in the algebraic geometry of its mirror. At the onset of mirror symmetry, these algebro-geometric computations were much easier and then they were then used as a guiding principle for what we aim to prove in symplectic geometry. This made century-old problems in enumerative geometry achievable. In 1994, the Fields Medallist Kontsevich provided a conjectural but fully mathematical version of mirror symmetry, encoding the symplectic geometry in what is called a Fukaya category and the algebraic geometry in a derived category of coherent sheaves. This provided a robust formulation in algebra of this physical and geometric phenomenon.Throughout the past three decades, mirror symmetry has expanded and it is now seen that mirror symmetry is not just a relationship amongst Calabi-Yau manifolds, but many more geometric spaces (e.g., Fano manifolds, log Calabi-Yau varieties). However, it has also been extended to the study of singularities. Interestingly, one can model the geometry of certain spaces by constructing a function so that the function is singular along the original space. Then one can deform this model and still obtain a physical model for string theory. This is an example of a Landau-Ginzburg model. Mirror symmetry has been established for Landau-Ginzburg models in a few cases, and it has been shown to be powerful in the study of classical higher-dimensional shapes such as Calabi-Yau manifolds. However, there are still foundational issues to be handled in the study of mirror symmetry for Landau-Ginzburg models. Ideally, we would like to prove a form of Kontsevich's conjecture for Landau-Ginzburg models, but before we do so in general, we will need to understand the algebro-geometric aspects of Landau-Ginzburg models. This project aims to better understand this categorical point of view for Landau-Ginzburg models, proving various structural results on their analogue of the derived category of coherent sheaves above, known as the (matrix) factorisation category.