Mirror Constructions: Develop, Unify, Apply

在这个项目中，我们研究受弦理论启发的几何问题。在弦理论中，我们将亚原子粒子视为弦，而不是点，这要求宇宙有六个额外的小维度，称为卡拉比-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。弦理论预言了惊人的数学，作为数学家，我们严格地证明了这一点。我们主要集中在研究可以被看作是一组多项式方程的解的形状。选择正确的数据，这样的方程组可以用来定义一个卡拉比-丘形状。弦理论预测了一种对偶性，即对于任何卡拉比-丘空间，都存在另一个称为镜像的空间。这两种形状之间的各种物理和几何数据交换，创造了一种被称为镜像对称的关系。该领域的一个关键问题是，给定Calabi-Yau空间，如何找到与之相关的镜像空间。一旦开发了显式构造，我们就可以检查镜像关系是否成立。文献中有各种各样的结构，有不同程度的镜像对称证据；然而，他们经常不同意！在这个项目中，我们的目标是处理这种差异，统一他们的方法。同样，我们的目标是创造新的Calabi-Yau品种，同时赋予它们镜像形状，增加目前存在的镜像对库。虽然Calabi-Yau空间通常很难可视化，但它们通常具有易于学习的代数描述。在这个项目中，我们经常会对Calabi-Yau形状进行很大的变形，以至于它不再是一个Calabi-Yau空间，而是一些更简单的代数结构，在物理文献中被称为Landau-Ginzburg模型。通过证明Landau-Ginzburg模型之间的关系，我们经常会发现Calabi-Yau形状之间的关系。因此，我们将能够以代数的方式将各种结构联系起来，以便更好地概述镜像建议。的确，这就解释了上述文献中不同的镜子结构之间的差异。此外，我们将研究与Landau-Ginzburg模型的代数关系，以建立新的法诺流形之间的关系。虽然有一个关于低维Fano流形分类的大型项目，但它们通常具有相同的有趣或内在的代数结构，称为（分数）Calabi-Yau范畴。我们的目标是运用我们的直觉，从统一的结构中找到这些基本数据之间的关系，以简化潜在的法诺流形之间的关系。最后，我们将我们对各种Calabi-Yau空间几何的理解应用于计算数论。卡拉比-丘形状的一维情况，即椭圆曲线，在过去几十年里在密码学中发挥了主导作用；然而，最近有一些建议导致需要对更高维度有更多的了解。通过与计算数论学家的互动，我们将分离出基本的Calabi-Yau形状，这些形状表现出有趣的显式数论现象，从而导致l系列的应用。

In this project, we research geometric problems inspired by string theory. In string theory, we view subatomic particles as strings, not points, requiring the universe to have six extra small dimensions called a Calabi-Yau shape. If we trace the string as it moves through time, it creates a (Riemann) surface. String theory has predicted amazing mathematics, which we, as mathematicians, prove rigorously.We are mainly focussed on studying shapes that can be viewed as the solution to a set of polynomial equations. Chosen with the correct data, such a system of equations can be used to define a Calabi-Yau shape. String theory predicts a duality that states that, for any Calabi-Yau space, there exists another space called the mirror. Various physical and geometric data between these two shapes is exchanged, creating a relationship that has come to be known as mirror symmetry. A key problem in this field is how one, given the Calabi-Yau space, finds the mirror space that is related to it. Once an explicit construction is developed, we then can check if a mirror relationship holds. There are various constructions in the literature with varying degrees of evidence of mirror symmetry; however, they often disagree! We aim in this project to deal with this discrepancy, unifying their approaches. In the same vein, we aim to potentially create new Calabi-Yau varieties while also giving their mirror shape, adding to the library of mirror pairs that currently exist.While Calabi-Yau spaces are often very difficult to visualize, they often have algebraic descriptions that are easy to study. In this project, we often will deform the Calabi-Yau shape so much that it is no longer even a Calabi-Yau space but some easier algebraic structure, known in the physics literature as a Landau-Ginzburg model. By proving relations between Landau-Ginzburg models, we will often find relations between Calabi-Yau shapes themselves. Thus, we will be able to relate various constructions algebraically in order to create a better overview of mirror proposals. Indeed, this explains the discrepancy above between different constructions for mirrors in the literature.In addition, we will study the algebraic relations to Landau-Ginzburg models in order to create new relations between Fano manifolds. While there is a large project regarding classification of Fano manifolds in low dimension, they often have the same interesting or intrinsic piece of algebraic structure, known as a (fractional) Calabi-Yau category. We aim to apply our intuition from unifying constructions in order to find relations between this fundamental data in order to streamline the relations between potential Fano manifolds. Lastly, we apply our understanding of the geometry of various Calabi-Yau spaces to computational number theory. The one-dimensional case of a Calabi-Yau shape, the elliptic curve, has played a leading role in cryptography in the last few decades; however, there have been recent proposals that have led to needing more understanding of higher dimensions. By interacting with computational number theorists, we will isolate fundamental Calabi-Yau shapes that exhibit interesting explicit number-theoretic phenomena, leading to applications for L-series.