Mirror symmetry and quiver flag varieties

Fano簇是几何学的基本组成部分之一。因为它们是如此基础，分类法诺品种是一个主要的开放问题，与应用到许多领域的数学（如。双有理几何，Calabi-Yau流形）。分类的法诺品种只知道到3维，这是一个主要的成果，20日世纪代数几何。在3维情况下使用的工具是不适合更高的维度，然而，一个有前途的方法的分类是通过镜像对称。镜像对称性是最初出现在弦理论中的一系列理论和定理;它是国际上活跃研究的一个快节奏领域。一个猜想是，每一个法诺簇对应于一个非常简单的对象：洛朗多项式。这个洛朗多项式被称为Fano簇的镜像伙伴。如果这些特征被证明，法诺变种可以通过它们的镜像伙伴进行分类，这是一个不那么困难的问题。某些简单法诺簇（复曲面簇）的镜像对称性是很好理解的：这项基础工作在数学的许多领域都产生了深远的影响。我的研究的主要目标是将这些结果推广到另一类更复杂的Fano品种，称为Flag品种。箭袋旗变种在Fano分类程序中作为Fano变种的环境空间是重要的。 我的主要策略研究镜像对称性的国旗品种是减少问题的国旗品种的问题复曲面品种，通过两种技术：阿贝尔/非阿贝尔对应和复曲面退化。 阿贝尔/非阿贝尔对应可以用来理解量子上同调。量子上同调是镜像对称的“A面”：镜像对称图给出了法诺簇的量子上同调和它们镜像伙伴的相应“B面”数据之间的对应关系。复曲面退化提供了一种研究镜像对称B面的方法。总之，这两种技术将使我能够证明镜像定理的旗帜品种：更基本的版本在短期内，在形式的洛朗多项式镜像，但在长期更强的版本，将开辟新的研究途径（如镜像对称卡拉比-丘子品种，法诺分类在这种情况下，舒伯特演算和镜像对称之间的新联系.）。 该提案的成果将加强加拿大与其他镜像对称研究中心之间的研究联系。从事这项工作的学生将接受新的数学研究所需技能的培训，并将使用计算机软件进行大规模的数学实验。他们将连接并为活跃的国际研究人员社区做出贡献，使他们能够很好地继续在学术界或工业界的职业生涯。

Fano varieties are one of the basic building blocks of geometry. Because they are so foundational, classifying Fano varieties is a major open problem, with applications to many areas of mathematics (eg. birational geometry, Calabi--Yau manifolds). The classification of Fano varieties is only known up to dimension 3; this was one of the major results of 20th century algebraic geometry. The tools used in the 3 dimensional case are not suitable to higher dimensions; however, a promising approach to the classification is via mirror symmetry. Mirror symmetry is a body of conjectures and theorems that originally arose in string theory; it is a fast paced area of international active research. One conjecture is that every Fano variety corresponds to a very simple object: a Laurent polynomial. This Laurent polynomial is called the mirror partner of the Fano variety. If these conjectures are proven, Fano varieties could be classified via their mirror partners, a much less difficult problem. Mirror symmetry for certain simple Fano varieties (toric varieties) is well understood: this foundational work has had profound impacts in many areas of mathematics. The main goal of my research is to generalize these results to another, more complicated, class of Fano varieties, called quiver flag varieties. Quiver flag varieties are important in the Fano classification program as ambient spaces of Fano varieties. My main strategy for studying mirror symmetry for quiver flag varieties is to reduce questions about quiver flag varieties to questions about toric varieties, via two techniques: the Abelian/non-Abelian correspondence and toric degenerations.  The Abelian/non-Abelian correspondence can be used to understand quantum cohomology. Quantum cohomology is the `A-side' of mirror symmetry: mirror symmetry conjectures give a correspondence between the quantum cohomology of Fano varieties and the corresponding `B-side' data of their mirror partners. Toric degenerations give a way of studying the B-side of mirror symmetry. Together,  these two techniques will allow me to prove mirror theorems for quiver flag varieties: more basic versions in the shorter term, in the form of Laurent polynomial mirrors, but in the longer term stronger versions that will open up new avenues of research (such as mirror symmetry for Calabi--Yau subvarieties, Fano classification in this context, new connections between Schubert calculus and mirror symmetry...).   The outcomes of this proposal will strengthen research connections between Canada and other centers of mirror symmetry research. Students who work on this proposal will be trained in the skills required for novel mathematical research, and will also use computer software to carry out mathematical experimentation on a large scale. They will be connected and contribute to an active international community of researchers, positioning them well to continue on to careers in either academia or industry.