Bridging Frameworks via Mirror Symmetry

站在一个地方。问这样一个问题：“当你站在那里时，你可能会遇到什么情况？”“一个答案是从0度到360度，但这并不是一个完全令人满意的答案。最直观的答案是你可以转个圈。这个答案是可能解的几何分类的一个例子，或者说是模空间。模空间在几何中无处不在。从圆锥曲线到机器人的运动范围，我们正在研究模空间。在代数几何中，我们研究多项式解的几何性质和相关的几何分类问题。当一个人有很多变量并使用更高的度时，这样的问题就变得困难了。通常有三种方法来研究变异：观察它们内部的其他物体，找到它们在其他物体内部的方式，找到帮助分类它们的不变量。在过去的25年里，弦理论为研究某些经典的代数几何对象（Calabi-Yau形状）提供了直观的框架。在弦理论中，为了得到宇宙的物理模型，将卡拉比-丘形状添加到时空连续体中。在数学中，这导致了一种称为镜像对称的几何对偶性，它关注的是IIA型弦理论和IIB型弦理论之间的对偶性。这个丰富的框架允许数学领域之间的许多联系，特别是辛几何和代数几何。许多联系都与枚举几何有关，研究某种类型的曲线在高维物体中有多少。镜像对称将辛几何中的这个问题变成了一个代数几何问题，使计算答案变得更容易。其中一些联系存在于数论中。变体有数论上的类似物，人们可以在有限的域上研究它们，提供黎曼ζ函数的几何类似物。提出的研究计划侧重于寻找由镜像对称驱动的领域之间的桥梁。本提案涉及以下项目：1.)提供了一种计算辛几何中fjrw不变量的方法，通过将不变量连接到代数几何设置，然后使用热带几何。这些不变量描述了某种类型的曲线有多少条位于Calabi-Yau形状的广义版本中，称为Landau-Ginzburg模型。研究镜面对称下Calabi-Yau形状的数论性质，利用与这些形状相关的zeta函数的性质。利用几何商对多边形的对应关系对某类高维类似物进行分类，从而给出某些类型的Calabi-Yau形状的分类。将镜像对称对另一种弦理论的意义编入法典，即异质镜像对称。这里提出的工作将在数学领域之间提供更多的联系，创造一个更有凝聚力的数学社区。每个项目都有两个领域，并以某种方式将它们联系起来，以便这两个领域都有助于对Calabi-Yau形状的理解。

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.