Homfly polynomials from the hilbert schemes of a planar curve: [after D. Maulik, A. Oblomkov, V. Shende,...]

Among the most interesting invariants one can associate with a link ${\mathcal L} \subset S^3$ is its HOMFLY polynomial $P({\mathcal L},v,s) \in {\mathbb Z}[v^{\pm1}, (s-s^{-1})^{\pm1}].$ A. Oblomkov and V. Shende conjectured that this polynomial can be expressed in algebraic geometric terms when $\mathcal L$ is obtained as the intersection of a plane curve singularity $(C,p) \subset {\mathbb C}^2$ with a small sphere centered at $p$: if $f =0$ is the local equation of $C$, its Hilbert scheme $C_p^{[n]}$ is the algebraic variety whose points are the length $n$ subschemes of $C$ supported at $p$, or, equivalently, the ideals $I \subset {\mathbb C}[[x,y]]$ containing $f$ and such that $\dim {\mathbb C}[[x,y]]/I=n$. If $m: C_p^{[n]} \to {\mathbb Z}$ is the function associating with the ideal $I$ the minimal number $m(I)$ of its generators, they conjecture that the generating function $Z(C,v,s)=\sum_n s^n \int_{C_p^{[n]}}(1-v^2)^{m(I)}d\chi(I)$ coincides, after a renormalization, with $P({\mathcal L},v,s)$. In the formula the integral is done with respect to the Euler characteristic measure . A more refined version of this surprising identity, involving a ``colored" variant of P(L,v,s), was conjectured to hold by E. Diaconescu, Z. Hua and Y. Soibelman. The seminar will illustrate the techniques used by D. Maulik to prove this conjecture.