Bridging Frameworks via Mirror Symmetry

站在一个地方。问这样一个问题：“当你站在那里的时候，你可能会面对什么方式？”一个答案是从零度到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绕圈转一圈。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。模空间在几何学中无处不在。从圆锥曲线到机器人的运动范围，人们正在研究模空间。在代数几何中，我们研究多项式的解的几何及其相关的几何分类问题。当一个人的变量很多，使用更高的学位时，这样的问题就变得困难了。这种形状通常有三种研究方法：观察位于它们内部的其他物体，找到它们位于其他物体内部的方法，以及找到有助于分类的不变量。在过去的25年里，弦理论为研究某些经典的代数几何物体提供了直观的框架，Calabi-Yau形状。在弦理论中，Calabi-Yau形状被添加到时空连续体中，以获得宇宙的物理模型。在数学上，这导致了一种被称为镜像对称的几何对偶，它侧重于IIA型和IIB型弦理论之间的对偶性。这个丰富的框架允许数学场之间的许多联系，通常是辛几何和代数几何。许多联系都与列举几何有关，研究高维对象中有多少特定类型的曲线。镜像对称性将辛几何中的这一问题转化为代数几何问题，使求解变得更容易。其中一些联系存在于数论中。簇具有数论类比，人们可以在有限域上研究它们，提供与Riemann Zeta函数的几何类比。拟议的研究计划侧重于寻找镜像对称性所激发的领域之间的桥梁。该建议涉及以下项目：1)提供了一种计算辛几何中FJRW不变量的方法，方法是将不变量连接到代数几何设置，然后使用热带几何。这些不变量描述了在Calabi-Yau形状的广义版本中有多少特定类型的曲线，称为Landau-Ginzburg模型。研究镜像对称下的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。通过几何商，利用代数对象与代数对象的对应关系，对某类高维类似于多边形进行分类，从而给出某些类型的Calabi-Yau形状的分类。编纂镜像对称对另一种类型的弦理论--杂化镜像对称的意义。这里提供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。每个项目都有两个场，并以某种方式将它们连接在一起，这样两个场都可以为理解Calabi-Yau形状做出贡献。

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.