Abstract: I will discuss algebraic structures associated to moduli of
sheaves on elliptic surfaces, and describe their relation with other
parts of mathematical physics. These algebraic structures control the
enumerative geometry of these moduli spaces analogous to how quantum
groups control enumerative invariants of quiver varieties. The main
results discussed will include a description of the quantum differential
equation in these geometries and work in progress describing the
relevant algebras as Hopf algebras with generalized versions of R matrices.