Quantum Difference Equations for Grassmannians

We consider the quantum difference equation (QDE) for the equivariant quantum K-theory of the Grassmannian. In this thesis we obtain a solution to the QDE and rely on results from Schubert calculus to prove our solution. Then we use the solution to derive the Bethe ansatz equations. In the limit, we obtain similar results for the cohomological analogue. However, in the cohomological case we depend on the Satake isomorphism for our proof. For both cases, we describe a basis-free form of the fundamental solution. For this, we derive Cauchy identities using vertex models. As an application, we identify the quantum K-theory ring of Gr(k,n) with a quantum 5-vertex XXZ integrable spin chain. We perform a similar application in the cohomology case, identifying the quantum cohomology ring of Gr(k,n) with the 5-vertex XXX integrable spin chain.