|Location:||MSRI: Simons Auditorium|
I will begin with a theorem from hep-th/9410188, which basically states that a certain monodromy of "iniversal" R-matix qKZ, a generalization of the I.Frenkel-Reshetikhin one, is an R-matrix too, and therefore can be used for a qKZ of the "next level". The main application was the construction of the elliptic-polynomial representation of DAHA (in Looijenga -type functions). It results in the difference-elliptic QMBP, generalizing the Ruijsenaars one. Its integration is related to the elliptic envelope and the corresponding wall-crossing. The "ultimate" elliptic-elliptic theory can be formally defined in the same way, but its convergence remains a challenge.
After the break, I can discuss 2 related theories. The first is the Harish-Chandra theory of "global" q,t-hypergeometric functions (Ch, Stokman). Their Whittaker limits t->0 are some generating function of affine Schubert cells, which sector expansions generalize the Givental-Lee function (with "wall crossing" between the sectors). The second is the theory of elliptic Hall functions,presumably the case of zero (second) central charge of the elliptic envelope (to be compared with Smirnov's formula), directly related to the DAHA-Satake map: arXiv:0904.4324. This will depend on the suggestions of the participants.