Open Mirror Geometry for Landau-Ginzburg Models

这个奖学金研究几何问题的弦论。在弦理论中，人们把亚原子粒子看作弦，而不是点，这就要求宇宙有六个额外的小维度，以所谓的卡-丘形状的形式存在。如果我们追踪弦在时间中的运动，它会创建一个（黎曼）曲面。弦理论指出了我们最初没有看到的数学结构。现在，数学在弦理论的指导下取得了巨大的进步，推动了几何学、代数学甚至弦理论本身的发展，弦理论对现代几何学的革命最明显的方式就是枚举几何学。枚举问题的一个例子是“在笛卡尔平面上有多少条直线通过两个给定的点？“答案是我们从中学就知道的：两点之间有一条独特的直线。我们现在问：“有多少黎曼曲面/弦（给定度或亏格）是卡-丘形状的？“这是一个经典的几何问题，可以以某种形式追溯到世纪。这些问题是希尔伯特在1900年提出的第15个问题的基础。像这样的枚举问题有助于我们理解高维空间，这是几何学家的一个关键问题。在这里，我们使用现代的思想来解决世纪的问题，同时也研究当代的变体。今天，我们通过使用弦理论中的对偶来计算黎曼曲面，将计数编码为多元积分。这种关系是一种称为镜像对称的场的表现。这种对偶使用不同类型的几何交换两个不同卡-丘形状的数据：（1）位于辛几何领域的枚举几何和（2）位于代数几何领域的多元积分。有这种数据交换的两个卡-丘图形被称为镜像。镜像对称在过去的三十年里一直是现代几何的关键催化剂之一，并且只会越来越有动力。镜像对称的一个关键问题是：给定一个卡-丘形状，我如何构造镜像？更广泛地说，人们可以问如何构造任何辛流形及其变形的镜像。在过去的10年里，人们一直在努力理解镜像对称如何适用于给定卡-丘形状的所有变形。粗略地说，当一个卡-丘形状变形太厉害时，最终得到的不再是一个空间，而是一个复值函数，称为朗道-金兹伯格模型。卡-丘形状的几何体现在被封装在这个函数中，在那里它更容易计算。2013年，量子奇异性理论（FJRW theory）提出了计算朗道-金兹伯格模型的黎曼曲面的类似理论。然而，目前还没有系统的方法来构造朗道-金兹伯格模型的镜像。本研究旨在解决朗道-金兹伯格模型的上述关键问题，提供一种直接构造朗道-金兹伯格模型镜像的方法。实际上，这将提供一个更“全球”的方法来构建镜像，使人们能够更有效地研究辛空间的变形。

This fellowship researches geometric problems made accessible by string theory. In string theory, one views subatomic particles as strings, not points, requiring the universe to have six extra small dimensions in the form of what is known as a Calabi-Yau shape. If we trace the string as it moves through time, it creates a (Riemann) surface. String theory has pointed out mathematical structure that we initially did not see. Now, mathematics informed by string theory has created great advances that has pushed the boundaries of geometry, algebra and even string theory itself.The most clear way in which string theory has revolutionised modern geometry is in enumerative geometry. An example of an enumerative problem is "How many lines in the cartesian plane go through two given points?" The answer is something we have known since secondary school: there's a unique straight line between two points. We now ask: "How many Riemann surfaces / strings (of a given degree or genus) are in a Calabi-Yau shape?" This is a classical problem in geometry, going back in some form to the 19th century. Such questions were the basis to Hilbert's 15th problem posed in 1900. Answering enumerative problems like this one helps us understand higher-dimensional spaces, a key problem for geometers. Here, we use modern ideas to tackle problems over a century old, while also studying contemporary variants.Today, we count Riemann surfaces by using dualities in string theory to encode the counts into multivariate integration. Such a relation is the manifestation of a field called mirror symmetry. This duality exchanges the data of two different Calabi-Yau shapes using different types of geometry: (1) the enumerative geometry that sits squarely in the field of symplectic geometry and (2) the multivariate integration which is placed in the field of algebraic geometry. The two Calabi-Yau shapes that have this exchange of data are called mirrors. Mirror symmetry has been one of the key catalysts of modern geometry for the past thirty years, and is only gaining momentum.A key question in mirror symmetry is: given a Calabi-Yau shape, how do I construct the mirror? More broadly, one can ask how to construct the mirror for any symplectic manifold and its deformations. In the past 10 years, there has been work in trying to understand how mirror symmetry works for all deformations of a given Calabi-Yau shape. Roughly speaking, when one deforms a Calabi-Yau shape too hard, one ends up with not a space anymore, but a complex-valued function known as a Landau-Ginzburg model. The geometry of the Calabi-Yau shape is now encapsulated in this function, where it is easier to compute. The analogous theory for counting Riemann surfaces for Landau-Ginzburg models, known as FJRW theory or quantum singularity theory, was developed in 2013; however, there is no systematic way in any large generality for how one can construct the mirror to a Landau-Ginzburg model.This fellowship aims to solve the key question above for Landau-Ginzburg models, providing a way to construct the mirror to a Landau-Ginzburg model directly. In effect, this will provide a more 'global' approach to constructing mirrors, allowing for one to study deformations of symplectic spaces more effectively.