Open Mirror Geometry for Landau-Ginzburg Models

该奖学金研究由弦理论解决的几何问题。在弦理论中，人们把亚原子粒子视为弦，而不是点，这就要求宇宙有6个额外的小维度，即所谓的卡拉比-丘形状。如果我们跟踪弦在时间中的运动，它会产生一个（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，以弦理论为基础的数学已经取得了巨大的进步，突破了几何、代数甚至弦理论本身的界限。弦理论彻底改变现代几何的最明显方式是在枚举几何中。枚举问题的一个例子是“笛卡尔平面上有多少条直线经过两个给定的点？”答案是我们从中学起就知道的：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（给定度或属）是Calabi-Yau形状？”这是一个经典的几何问题，以某种形式可以追溯到19世纪。这些问题是希尔伯特1900年提出的第15个问题的基础。回答像这样的列举性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，我们用现代的思想来解决一个世纪以前的问题，同时也研究当代的变体。今天，我们利用弦理论中的对偶性对黎曼曲面进行计数，将计数编码为多元积分。这种关系是镜像对称领域的表现。这种对偶性使用不同类型的几何来交换两种不同的Calabi-Yau形状的数据：(1)恰好位于辛几何领域的枚举几何；(2)位于代数几何领域的多元积分。具有这种数据交换的两个Calabi-Yau形状被称为镜像。在过去的三十年里，镜像对称一直是现代几何的关键催化剂之一，而且势头还在不断增强。镜面对称的一个关键问题是：给定一个卡拉比-丘形状，我如何构造镜面？更广泛地说，人们可以问如何为任何辛流形及其变形构造镜像。在过去的10年里，人们一直在努力了解镜面对称如何在给定的卡拉比-丘形状的所有变形中起作用。粗略地说，当一个人把一个Calabi-Yau形状变形得太厉害时，他最终得到的不再是一个空间，而是一个被称为朗道-金兹堡模型的复值函数。Calabi-Yau形状的几何形状现在被封装在这个函数中，这样更容易计算。朗道-金兹堡模型中黎曼曲面计数的类似理论，被称为FJRW理论或量子奇点理论，于2013年提出；然而，对于如何构建朗道-金兹堡模型的镜像，在任何大的一般性中都没有系统的方法。本研究旨在解决上述Landau-Ginzburg模型的关键问题，为直接构建Landau-Ginzburg模型的镜像提供一种方法。实际上，这将提供一种更“全局”的方法来构造镜子，允许人们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.