Ivan Cherednik (UNC Chapel Hill) will give two lectures at ETH.
Wednesday November 12, at 13:15 in Room HG G 43, ETH
DAHA superpolynomials of iterated torus knots
Abstract: The DAHA-Jones polynomials (refined, with an extra parameter)
of iterated torus knots, including all algebraic knots, will be defined for any root systems and weights. Conjecturally, they generalize the Jones-WRT invariants based on Quantum Groups, which was checked for torus knots in types A-C. For A_n, the DAHA-Jones polynomials can be combined in one superpolynomial, presumably coinciding with the stable Khovanov-Rozansky one for sl(n+1) of the corresponding knot and superpolynomials via the M5 theory (the BPS states, String Theory); the latter approach is not mathematically rigorous. The DAHA-superpolynomials match the superpolynomials obtained from perfect modules of rational DAHA (Gorsky, Oblomkov, Rasmussen, Shende), though using rational DAHA is restricted to the torus knots and (so far) only to the unclored polynomials. Conjecturally, DAHA provides the formulas for Betti numbers of Jacobian factors of plane singularities, which are very difficult to calculate in algebraic geometry. This is a great test of the maturity of the new theory, closely related to the Oblomkov-Rasmussen-Shende conjecture.
Thursday November 13, at 15:15 in Room HG G 43, ETH
Global hypergeometric and Whittaker functions
Abstract: The lecture will be devoted to the new theory of global difference hypergeometric and Whittaker functions, one of the major applications of the double affine Hecke algebras and a breakthrough in the classical harmonic analysis. They integrate the Ruijsenaars-Macdonald difference QMBP and "q-Toda" (any root systems), and are analytic everywhere ("global") with superb asymptotic behavior. The definition of the global functions was suggested about 14 years ago; it is conceptually different from the definition Heine gave in 1846, which remained unchanged and unchallenged since then. The analytic theory of these functions was completed only recently (the speaker and J. Stokman). If time permits, the connection of the Harish-Chandra theory of global q-Whittaker functions will be discussed with the Givental-Lee formula (Gromov-Witten K-theoretic invariants of flag varieties) and that due to Braverman and Finkelberg (affine flag varieties).