Mirror symmetry for flag varieties

Algebra and geometry come together when studying the solution sets of polynomial equations in many variables -- so-called algebraic varieties. My research centers on flag varieties, which are particularly beautiful algebraic varieties with a very rigid structure. For example they are 'homogeneous': They have a large (matrix) symmetry group which can translate any point into any other. Flag varieties come in series starting with the simplest example, the Riemann sphere, and reaching arbitrarily high dimension. Mirror symmetry came to the attention of mathematicians in the early 1990's when physicists made astounding and precise predictions about certain 3-dimensional algebraic varieties and numbers of rational curves on them. Since then the process of unraveling what underlies these predictions has been progressing, and the field has become a major part of modern mathematics. In my research I propose to study mirror symmetry in the context of flag varieties. There has already been a great deal of work on the theory of quantum cohomology for flag varieties which arose out of mirror symmetry and is a very rich subject in its own right. But this research has been almost entirely from a classical perspective of quantum cohomology generalizing ordinary cohomology, which has not involved mirror symmetry at all. What has been left out is the mysterious mirror model, what the physicists used in their computations of numbers of rational curves, which holds the information about enumerative algebraic geometry in a completely different form. In my recent paper [14] I propose such a model explicitly for general flag varieties and relate it to quantum cohomology. The main goal of the proposed research is to use this mirror model to understand the deeper and more difficult Gromov-Witten theory for flag varieties.