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Plane Curve Singularities via DAHA

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Ivan Cherednik
UNC Chapel Hill, RIMS
Fri, 2017-06-23 11:00 - 12:00
MPIM Lecture Hall

Double Affine Hecke Algebras provide a universal approach to the action of $PSL(2,\mathbb{Z})$ in algebra, analysis and physics. So it is not too surprising that they appeared useful in low-dimensional geometry. The main applications now are in the theory of refined (with extra parameters) Jones, WRT, HOMFLY-PT polynomials of iterated links, including all algebraic ones. The DAHA-Jones polynomials and DAHA-superpolynomials will be defined.

The DAHA-superpolynomials of iterated knots conjecturally coincide with their stable Khovanov-Rozansky polynomials. The latter are involved apart from the Khovanov polynomials. The following conjectural relation is more verifiable. The DAHA-superpolynomials can be interpreted in terms of the compactified Jacobians of the corresponding unibranch plane curve singularities. Accordingly they generalize the $p$-adic orbital integrals (in type A). This includes a presentation of Puiseux exponents of such singularities as proper sequences of $PSL(2,\mathbb{Z})$-matrices (then DAHA is used). An immediate application of this conjecture is that the orbital integrals are topological invariants of singularities. Interestingly, this construction is parallel to that for the periods of cusp forms, which I hope to discuss a bit if time permits.

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