The Hilbert scheme of a plane curve singularity and the HOMFLY
polynomial of its link -- Vivek Shende, March 25, 2010

<p>

The intersection of a complex plane curve with a
small three-sphere surrounding one of its singularities is a
non-trivial link. The refined punctual Hilbert schemes of the
singularity fixed length and whose defining ideals have a fixed
number of generators.  I will describe a conjecture equating
the generating function of Euler characteristics of refined
punctual Hilbert schemes to the HOMFLY polynomial of the link.
Some evidence will be provided, including a proof in the
unibranch "toric case," i.e., for singularities of the form x^p
= y^q.  This talk presents joint work with Alexei Oblomkov.