Superpolynomials of plane curve singularities and zeta-functions

2018/05/09 水 16:30 - 17:30
Ivan Cherednik
University of North Carolina at Chapel Hill

The talk will be about a recent surprising development in the highly interdisciplinary theory of superpolynomials of plane curve singularities, connecting them with the zeta-functions of the corresponding rings over finite fields. The superpolynomials are stable Khovanov-Rozansky polynomials of algebraic links, which conjecturally coincide with the DAHA superpolynomials and those describing the BPS states in string theory. They are also directly related to affine Springer fibers, p-adic orbital integrals and Hilbert schemes of plane curve singularities and the complex plane. The zeta-functions are essentially due to Galkin and Stohr (though we will need some generalization), a very classical direction in number theory. Their definition and examples will be provided and we will formulate a positivity conjecture connecting them with Jacobian factors of plane curve singularities (local factors of the compactifeid Jacobians).