Bridging Frameworks via Mirror Symmetry

通过镜像对称桥接框架

• 批准号: EP/N004922/1
• 负责人: Tyler Kelly
• 金额: $ 28.37万
• 依托单位: University of Cambridge
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2015
• 资助国家: 英国
• 起止时间: 2015 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FN004922%2F1
• 关键词: Bridging; Frameworks; via; Mirror; Symmetry; Bridging; Frameworks; via; Mirror; Symmetry

Stand in one place. Ask the question "What are the possible ways you could face while standing there?'' One answer is from zero degrees to 360 degrees, but that is not a fully-satisfying answer. The most intuitive answer is you can turn around in a circle. This answer is an example of a geometric classification of possible solutions, or a moduli space. Moduli spaces are ubiquitous in geometry. From conic sections to the range of motion of a robot, one is studying moduli spaces. In algebraic geometry, we study the geometry of the solutions of polynomials and associated geometric classification problems. When one has many variables and uses higher degrees, such questions become difficult. Such shapes formed by Typically there are three ways to study varieties: looking at other objects that sit inside them, finding ways that they sit inside other objects, and finding invariants that help classify them.In the last 25 years, string theory has giving intuitive frameworks for studying certain classical algebro-geometric objects, Calabi-Yau shapes. In string theory, Calabi-Yau shapes are added to the space-time continuum in order to get physical models for the universe. In mathematics, this led to a geometric duality called mirror symmetry which focuses on the duality between Type IIA and IIB string theory. This rich framework allows many connections between mathematical fields, typically symplectic geometry and algebraic geometry.Many of the connections made have to do with enumerative geometry, studying how many curves of a certain type sit inside higher dimensional objects. Mirror symmetry turned this problem in symplectic geometry into an algebro-geometric problem, making it easier to compute the answer. Some of the connections sit in number theory. Varieties have number-theoretic analogues where one can study them over a finite field, providing geometric analogues to the Riemann zeta function. The proposed research plan focuses on finding bridges amongst fields motivated by mirror symmetry. The proposal involves the following projects:1.) Providing a method to compute the FJRW-invariants in symplectic geometry by linking the invariants to an algebro-geometric setting then using tropical geometry. These invariants describe how many curves of a certain type sit in a generalized version of a Calabi-Yau shape, called a Landau-Ginzburg model.2.) Studying the number theoretic properties of Calabi-Yau shapes when viewed under mirror symmetry, harnessing properties of the zeta function associated to these shapes.3.) Classify a certain class of higher-dimensional analogues to polygons by using their correspondence to algebraic objects by using geometric quotients, consequently giving a classification of certain types of Calabi-Yau shapes.4.) Codify what mirror symmetry means for another type of string theory, heterotic mirror symmetry.The work presented here will provide more links amongst mathematical fields, creating a more cohesive mathematical community. Each project takes two fields and connects them in a way so that both fields can contribute to the understanding of Calabi-Yau shapes.

站在一个地方。询问问题：“站在那里时您可能面临什么可能面临的方式？”一个答案是从零度到360度，但这不是一个完全满意的答案。最直观的答案是，您可以圈出一个圈子。这个答案是可能的几何范围的示例，是一个可能的溶液或模态空间的一个机器人。一个正在研究代数的几何形状，我们研究多项式的几何形状和相关的几何分类问题。 25年，弦理论提供了直观的框架，用于研究某些经典的代数几何对象，calabi-yau形状。在弦理论中，为了获得宇宙的物理模型，添加了卡拉比YAU的形状。在数学中，这导致了一种几何二元性，称为镜像对称性，重点是IIA和IIA型字符串理论之间的二元性。这个丰富的框架允许数学字段之间的许多连接，通常是符号几何形状和代数几何。镜像对称性在符号几何形状中将这个问题转变为代数几何问题，从而更容易计算答案。一些连接位于数字理论中。品种具有数量理论类似物，可以在有限的领域进行研究，从而为Riemann Zeta函数提供几何类似物。拟议的研究计划着重于在镜子对称性动机的田地之间寻找桥梁。该提案涉及以下项目：1。）提供一种方法，通过将不变性链接到代数几何设置然后使用热带几何形状来计算符号几何形状中的fjrw-invariants。这些不变的人描述了某种类型的calabi-yau形状的一定曲线，称为landau-ginzburg型号。代数对象通过使用几何标的对象，从而对某些类型的卡拉比（Calabi-yau）形状进行分类。4。编写镜子对称对另一种类型的弦乐理论的含义，异性镜对称性。此处介绍的工作将在数学领域中提供更多的链接，从而创建一个更具凝聚力的数学社区。每个项目都采用两个字段并以某种方式将它们连接起来，以便两个字段都可以有助于对Calabi-yau形状的理解。