Abstract or Additional Information
Flag varieties are often studied by decomposing them into orbits of various special subgroups. This principle is also fruitful in the case of the affine flag variety, and I will explain a combinatorial tool for visualizing certain orbits inside the complete affine flag variety, originally due to Parkinson, Ram, and C. Schwer, now available in much greater generality via joint work with Naqvi, P. Schwer, and Thomas. As a concrete application, I will describe joint work with Arun Ram which provides a labeling of the points of the moduli space of genus zero curves in the complete complex flag variety using this combinatorial machinery of alcove walks. Following Peterson, this geometric labeling partially explains the "quantum equals affine" phenomenon which relates the quantum cohomology of this flag variety to the homology of the affine Grassmannian.