From Representations of Affine Algebras to the Quantum K-theory of Flag Manifolds and Combinatorics

I start by explaining a new twist of the classical connections between Lie algebra representations, flag manifolds $G/B$, and combinatorics. This relates certain modules over quantum affine algebras, the equivariant $K$-theory of semi-infinite flag manifolds, and the equivariant quantum $K$-theory of $G/B$. Based on these connections, my collaborators and I derived combinatorial multiplication formulas in $K$-theory and quantum $K$-theory, which are expressed in terms of the so-called quantum alcove model. These formulas led us to many applications, including solutions to several longstanding conjectures. The talk will be largely self-contained.