1 videos

5 videos

7 videos

5 videos

# Collection Bourbaki - Mars 2019

Organisateur(s)
Date(s) 30/03/2019 - 30/03/2019
URL associée https://www.bourbaki.fr/seminaires/2019/Prog_mar19.html
00:00:00 / 00:00:00
2 3

## [1160] HOMFLY polynomials from the Hilbert schemes of a planar curve

Among the most interesting invariants one can associate with a link $\mathcal L\subset S^3$ is its HOMFLY polynomial $\mathbf P(\mathcal L,v,s)\in \mathbf Z[v^{\pm 1},(s−s^{−1})^{\pm 1}$. 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 $(C,p)\subset \mathbf C^2$ 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 $I\subset C[[x,y]]$ containing $f$ and such that $\dim C[[x,y]]/I=n$. If $m:C_p^{[n]}\rightarrow\mathbf Z$ is the function associating with the ideal $I$ the minimal number $m(I)$ of its generators, they conjecture that the generating function $$Z(C,v,s)=\sum_n s^{2n}\int_{C_p^{[n]}}(1−v^2)^{m(I)}d\chi(I)$$ coincides, up to a renormalization, with $\mathbf P(\mathcal L,v,s)$. In the formula the integral is done with respect to the Euler characteristic measure $d\chi$. A more refined version of this surprising identity, involving a colored variant of $\mathbf P(\mathcal L,v,s)$, 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.

[After D. Maulik, A. Oblomkov, V. Shende...]

### Bibliographie

Séminaire Bourbaki, 71ème année (2018-2019), n°1160, mars 2019 PDF

### Dernières questions liées sur MathOverflow

Pour poser une question, votre compte Carmin.tv doit être connecté à mathoverflow

### Poser une question sur MathOverflow

• 01:09:49
publiée le 3 mars 2019

## [1159] The Riemann zeta function in short intervals

01:06:00
publiée le 3 mars 2019

## [1160] HOMFLY polynomials from the Hilbert schemes of a planar curve

De Luca Migliorini

01:00:41
publiée le 30 mars 2019

## [1161] Higher rank Teichmüller theories

De Beatrice Pozzetti

## Inscrivez-vous

• Mettez des vidéos en favori
• Ajoutez des vidéos à regarder plus tard &
conservez votre historique de consultation
• Commentez avec la communauté
scientifique
• Recevez des notifications de mise à jour
de vos sujets favoris
Donner son avis