Quantum cohomologies and the abelian-nonabelian correspondence

Quantum cohomology and Gromov-Witten theory have become active areas of research since their introduction in the early 1990s. In this dissertation two problems are explored. The first is a proof by localization of a certain case of the abelian-nonabelian correspondence [1, 2, 3]. The method is inspired by Kontsevich's computations of Gromov-Witten invariants by localization, resulting in combinatorial computations involving sums over trees. The second problem is that of finding closed forms for K-theoretic J-functions of homogeneous spaces. Closed forms are established for flag varieties of type A and formulas are proposed for flag varieties of other classical types.