Mirkovic, IvanDong, Zhijie2024-04-262024-04-262018-092018-0910.7275/12715994https://hdl.handle.net/20.500.14394/17542The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian to a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. The relation between this conjecture and the reduceness conjecture is explained at the end.Loop GrassmannianMV cyclepreprojective algebraquiver GrassmannianAlgebraDissertation (Open Access)