Mirror symmetry and quiver flag varieties

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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.