Mirror Constructions: Develop, Unify, Apply

在这个项目中，我们研究了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，这需要宇宙有六个额外的小维度，称为卡拉比-尤形状。如果我们追踪弦在时间中的移动，它就会产生一个(黎曼)表面。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。我们主要专注于研究可以被视为一组多项式方程的解的形状。选择了正确的数据，这样的方程系统就可以用来定义卡拉比-尤形状。弦理论预言了一个二元性，即对于任何Calabi-Yau空间，都存在另一个称为镜像的空间。这两个形状之间交换了各种物理和几何数据，创建了一种称为镜像对称的关系。这个领域的一个关键问题是，在给定Calabi-Yau空间的情况下，如何找到与其相关的镜像空间。一旦显式结构被开发出来，我们就可以检查镜像关系是否成立。在文献中有不同程度的镜像对称的证据；然而，他们经常不同意！我们在这个项目中的目标是处理这种差异，统一他们的方法。同样，我们的目标是潜在地创造新的Calabi-Yau变种，同时给出它们的镜像形状，增加现有的镜像对库。虽然Calabi-Yau空间通常很难可视化，但它们通常有易于研究的代数描述。在这个项目中，我们经常会将Calabi-Yau形状变形得如此之多，以至于它甚至不再是Calabi-Yau空间，而是一些更容易的代数结构，在物理学文献中被称为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形状本身之间的关系。因此，我们将能够以代数方式将各种结构联系起来，以便更好地概述镜像建议。事实上，这解释了文献中不同镜面结构之间的差异。此外，我们还将研究Landau-Ginzburg模型的代数关系，以便在Fano流形之间建立新的关系。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固有的代数结构，称为(分数)Calabi-Yau范畴。我们的目标是从统一结构中应用我们的直觉，以便找到这些基本数据之间的关系，以便简化潜在Fano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到计算数论中。在过去的几十年里，Calabi-Yau形状的一维椭圆曲线在密码学中发挥了主导作用；然而，最近的一些提议导致了对更高维的更多理解。通过与计算数论者的互动，我们将分离出表现出有趣的显式数论现象的基本Calabi-Yau形状，从而导致L级数的应用。

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.