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In this talk, we will discuss an open Gromov-Witten invariant on hyperKahler surfaces, including K3 surfaces and certain Hitchin moduli spaces. The invariant is defined via the Lagrangian Floer theory and satisfy the Kontsevich-Soibelman wall-crossing formula and are expect to recover the generalized Donaldson-Thomas invariants studied in the work of Gaiotto-Moore-Neitzke.

The notion of Positive Representations is a new research program devoted to the representation theory of split real quantum groups, initiated in a joint work with Igor Frenkel. It is a generalization of the special class of representations considered by J. Teschner for Uq(sl(2,R)) in Liouville theory, where it exhibits a strong parallel to the finite-dimensional representation theory of compact quantum groups, but at the same time also serves some new properties that are not available in the compact case. In this talk, I will survey the recent developments and describe some of its relations to other areas of mathematics. 

I will discuss some results on double loop groups that point to geometric phenomena about double affine flag varieties and double affine Grassmannian. One result of this study is a definition of double affine Kazhdan-Lusztig polynomials.