Mirror symmetry and quiver flag varieties

范诺变体是几何的基本组成部分之一。因为它们是如此的基础，分类法诺变量是一个主要的开放问题，应用于数学的许多领域。两族几何（Calabi—Yau流形）。Fano品种的分类只知道维度3；这是20世纪代数几何的主要成果之一。三维情况下使用的工具不适用于高维；然而，一种很有前途的分类方法是通过镜像对称。镜像对称是最初出现在弦理论中的一系列猜想和定理；这是一个快节奏的国际活跃研究领域。一个猜想是，每一个法诺变量都对应一个非常简单的对象：一个洛朗多项式。这个洛朗多项式被称为法诺变换的镜像伙伴。如果这些猜想得到证实，范诺品种就可以通过它们的镜像伙伴来分类，这是一个容易得多的问题。某些简单法诺变种（环面变种）的镜像对称已经被很好地理解：这一基础性工作在数学的许多领域产生了深远的影响。我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，称为颤旗品种。颤旗品种作为法诺品种的环境空间，在法诺分类程序中占有重要地位。我研究颤旗变体镜像对称性的主要策略是通过阿贝尔/非阿贝尔对应和环面退化两种技术，将颤旗变体的问题简化为环面变体的问题。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”：镜像对称猜想给出了Fano变体的量子上同调与其镜像伙伴对应的“b面”数据之间的对应关系。环面退化为研究镜像对称的b面提供了一种方法。总之，这两种技术将使我能够证明箭旗变种的镜像定理：短期内更基本的版本，以劳伦多项式镜像的形式，但从长远来看，更强大的版本将开辟新的研究途径（如卡拉比-丘亚种的镜像对称，在这种情况下的法诺分类，舒伯特微积分和镜像对称之间的新联系……）。这项提案的成果将加强加拿大和其他镜像对称研究中心之间的研究联系。参与这一提案的学生将接受新颖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.