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Abstract:
I will talk about a proof of the rational shuffle conjecture, which relates certain weighted counts of Dyck paths with labelings to certain natural polynomials arising from Macdonal polynomials and elliptic Hall algebras. In an earlier work of Erik Carlsson and myself the "usual" shuffle conjecture was proved with the help of a certain algebra of operators acting on symmetric functions, a close relative of DAHA and EHA. A new ingredient of the present work is a geometric interpretation of this algebra using torus braids. Then certain transformations of braids imply otherwise strangely looking relations in the algebra, and lead to a proof of the rational conjecture. In the end I will explain how this method may lead to a computation of Khovanov homology of torus links.