Quantum groups and their applications

Quantum groups were introduced independently by Drinfeld and Jimbo around 1985. Representation theory of quantum groups has been developed intensively in the past decades by many famous mathematicians. Many problems in representation theory of quantum groups have been solved but many fundamental problems are still open. In this project, we study representations of quantum groups and its connection with Grassmannian cluster algebras, Grassmannian cluster categories and scattering amplitudes in physics. We consider the following problems. 1. Study geometric cactus group actions on geometric crystals. 2. In BCFG types, define an algebra homomorphism D from the torus containing q-characters of simple modules of quantum affine algebras, and apply D to study relation between q-characters of representations of quantum affine algebras and characters of representations of KLR algebras. 3. Study a tropical version of braid group actions and Marsh-Scott twists on Grassmannian cluster algebras C[Gr(k,n)] in terms of g-vectors and tableaux. Compare Fraser`s braid group actions and Kashiwara, Kim, Oh, and Park`s braid group actions. 4. Make a connection between additive categorifications of Grassmannian cluster algebras and monoidal categorification of Grassmannian cluster algebras using g-vectors. Try to prove that rigid indecomposable modules (resp. real prime modules) are cluster variables using Iyama-Yoshino reduction. 5. Applications to physics: study factorizations of Grassmannian string integrals.