# Séminaire Betty B.

J'ai créé ce séminaire en pensant aux mathématicien·ne·s, et notamment aux plus jeunes. J'y demande à des collègues de présenter le contexte mathématique de certains exposés du Séminaire de mon aïeul, N. Bourbaki, pour les rendre plus accessibles ; ils pourront aussi en introduire quelques outils ou des motivations plus lointaines. — Betty B., Nancago, Janvier 2018.

# Séminaire N. Bourbaki

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Let $\Gamma$ be the fundamental group of a compact surface $S$ with negative Euler characteristic, and $G$~denote $\mathrm{PSL}(2,\mathbf{R})$, the group of isometries of the hyperbolic plane. Goldman observed that the Teichmüller space, the parameter space of marked complex structures on~$S$ can be identified with a connected component of the character variety $\mathrm{Hom}(\Gamma,G)/G$, which can be selected by means of a characteristic invariant. Thanks to the work of Labourie, Burger-Iozzi-Wienhard, Fock-Goncharov, Guichard-Wienhard we now know that, surprisingly, this is a much more general phenomenon: there are many higher rank semisimple Lie groups~$G$ admitting components of the character variety only consisting of injective homomorphisms with discrete image, the socalled higher Teichmüller theories. The richness of these theories is partially due to the fact that, as for the Teichmüller space, truly different techniques can be used to study them: bounded cohomology, Higgs bundles, positivity, harmonic maps, incidence structures, geodesic currents, real algebraic geometry... In my talk I will overview a number of recent results in the field (following Labourie, Burger-Iozzi-Wienhard, Fock-Goncharov, Guichard-Wienhard, Bonahon-Dreyer, Li, Zhang, Martone-Zhang, Bargaglia, Alessandrini-Li, Collier-Tholozan-Toulisse.)

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 $Z(C,v,s)=\sum_n s^{2n} \int_{C_p^{[n]}}(1-v^2)^{m(I)}d\chi(I)$ coincides, up to a renormalization, with $P({\mathcal L},v,s)$. 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 $P({\mathcal L},v,s)$, 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.

A classical idea for studying the behaviour of complicated functions, like the Riemann zeta function $\zeta(s)$, is to investigate averages of them. For example, the integrals over $T \leq t \leq 2T$ of various powers of $\zeta(1/2+it)$, sometimes multiplied by some other cleverly chosen function, have been investigated extensively to deduce upper and lower bounds for the maximum size of $\zeta(1/2+it)$. More recently, Fyodorov and Keating have proposed the investigation of much shorter integrals over $T \leq t \leq T+1$. This turns out to lead to interesting connections between various issues in number theory, analysis, mathematical physics and probability, such as branching random walk and multiplicative chaos. I will try to explain some of these connections, ideas from the proofs, and what they tell us about the zeta function.

### Brochure

Des brochures contenant les quatre exposés du Séminaire N. Bourbaki seront distribuées au début de chaque séance ; 300 exemplaires seront disponibles au cours de cette session.

Pour recevoir à l'avance le programme et les résumés de chaque séminaire, veuillez vous abonner en envoyant un mail à sympa@lists.ens.fr, en indiquant :
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### Remerciements

Une subvention du CNRS couvre une partie des frais d'organisation de ce Séminaire.

Association des collaborateurs de Nicolas Bourbaki
École normale supérieure
45 rue d'Ulm
F-75230 Paris cedex 05, FRANCE
Courriel : bourbaki@dma.ens.fr