Topology Seminar: Yongbin Ruan.

In early 90's, there are two mathematical theories motivated by physics, Verlinde theory counting the number of generalized theta function on a curve and quantum cohomology counting the number of holomorphic maps. Early explicit computation shows that the level $l$ $GL_n$ Verlinde algebra is isomorphic to quantum cohomology
ring of Grassmanian $G(n, n+l)$. In 1993, Witten gave a conceptual explanation of this isomorphism, by proposing an equivalence between the quantum field theories which govern the level-l GL Verlinde algebra and the quantum cohomology of the Grassmannian. His physical derivation of the equivalence naturally leads to a mathematical problem: these two objects are conceptually isomorphic (without referring to detailed computations). In the talk, I will explain a K-theoretic version of Witten's conjecture motivated by our observation that Verlinde invariants is a K-theoretic invariant and hence should be compared to the quantum K-theory of
Grassmanian. The main content of the talk is a reformation of the conjecture as the consequence of wall-crossing problem of GLSM and the resolution of the conjecture in rank 2. This is a joint work with Ming Zhang.