En utilisant la méthode du double quantique, nous construisons une $R$-matrice universelle pour la quantification de la superalgèbre de Lie $D(2,1,x)$. Nous utilisons ce résultat pour construire un invariant d’entrelacs et nous montrons qu’il est égal à une spécialisation du polynôme de Dubrovnik introduit par Kauffman.

A new algebraic structure on the orbits of dressing transformations of the quasitriangular Poisson Lie groups is provided. This gives the topological interpretation of the link invariants associated with the Weinstein-Xu classical solutions of the quantum Yang-Baxter equation. Some applications to the three-dimensional topological quantum field theories are discussed.

Let $M$ be a 1-connected closed manifold of dimension $m$ and $LM$ be the space of free loops on $M$. M.Chas and D.Sullivan defined a structure of BV-algebra on the singular homology of $LM$, $H_{*}(LM;k)$. When the ring of coefficients is a field of characteristic zero, we prove that there exists a BV-algebra structure on the Hochschild cohomology $H{H}^{*}(C^{*}\left(M\right);C^{*}\left(M\right))$ which extends the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between $H{H}^{*}(C^{*}\left(M\right);C^{*}\left(M\right))$ and the shifted homology $H_{*+m}(LM;k)$. We also prove that the...

We use the computational power of rational homotopy theory to provide an explicit cochain model for the loop product and the string bracket of a simply connected closed manifold $M$. We prove that the loop homology of $M$ is isomorphic to the Hochschild cohomology of the cochain algebra $C^{*}\left(M\right)$ with coefficients in $C^{*}\left(M\right)$. Some explicit computations of the loop product and the string bracket are given.

We give a survey of results on the Lieb-Thirring inequalities for the eigenvalue moments of Schrödinger operators. In particular, we discuss the optimal values of the constants therein for higher dimensions. We elaborate on certain generalisations and some open problems as well.

In this paper, we review some of our recent results in the study of the dynamics of interacting Bose gases in the Gross-Pitaevskii (GP) limit. Our investigations focus on the well-posedness of the associated Cauchy problem for the infinite particle system described by the GP hierarchy.

We define the Bloch spectrum of a quantum graph to be the map that assigns to each element in the deRham cohomology the spectrum of an associated magnetic Schrödinger operator. We show that the Bloch spectrum determines the Albanese torus, the block structure and the planarity of the graph. It determines a geometric dual of a planar graph. This enables us to show that the Bloch spectrum indentifies and completely determines planar $3$-connected quantum graphs.

The problem of recovering the singularities of a potential from backscattering data is studied. Let $\Omega$ be a smooth precompact domain in ${ℝ}^n$ which is convex (or normally accessible). Suppose $V_i=v+w_i$ with $v\in C_c^{\infty }({ℝ}^n)$ and $w_i$ conormal to the boundary of $\Omega$ and supported inside $\bar{\Omega }$ then if the backscattering data of $V_1$ and $V_2$ are equal up to smoothing, we show that $w_1-w_2$ is smooth.