** Mina Aganagic, UC Berkeley
**

Title: Two mathematical applications of (little) string theory

Abstract: I will describe two mathematical applications of a six-dimensional .little. string theory. The first leads to a precise relation between K-theoretic counts of instantons and q-deformed conformal blocks of W-algebras. The second application leads to a deformation of geometric Langlands correspondence. The geometric Langlands correspondence can be phrased in the language of conformal field theory. Its quantum q-deformation relates q-deformed conformal blocks of an affine Lie algebra and a W-algebra, associated to a Langlands dual pair of Lie groups. The correspondence can be proven in the simply laced case. The proof involves, in a crucial way, quantum K-theory of Nakajima quiver varieties, and the recently discovered elliptic stable envelopes. This is based on joint works with Nathan Haouzi and with Edward Frenkel and Andrei Okounkov.