Topological insulators are materials with very interesting properties from a physical point of view: they are electrical insulators in the bulk, while on the edge they can be characterised by the presence of stable currents. These properties are specific to the material's 'topological phase', i.e. its equivalence class with respect to continuous deformations in a given topology. This means that small external perturbations or microscopic irregularities in the material will not change the intrinsic physical properties of the phase.
In this talk, I will introduce the elements of solid state physics needed to describe these materials with exotic properties and the mathematical problems that emerge in this area. We will see how $K$-theory for $C^*$-algebras turns out to be a very useful and natural tool (in particular in a particular version, that of Van Daele) to classify topological insulators. Finally, I will present my work comparing two possible approaches for the classification and how the more concrete approach among them can be seen as a special case of the more abstract approach.
This presentation aims to present my PhD work about a generalization of gauge theory involving mathematical objects called supergroups, which represent a generalization of symmetry groups that originates in the physical ideas of supersymmetry.
I will first give an introduction to the usual Yang-Mills theory and the moduli spaces of instantons, in particular the ADHM construction which solves the moduli problem for a flat spacetime. Then, I will introduce the elements of supergeometry needed to formulate the super-moduli problem. Finally, I will present my recent research on the representability of the super-moduli functor.
J'expliquerai comment la structure géométrique et algébrique quantique émerge de l'espace des modules de théorie de jauge. Je discuterai également du rôle de l'intégrabilité quantique dans ce contexte et démontre comment on peut obtenir une famille nouvelle de systèmes intégrables quantiques.
We discuss few very recent results of works in progress (in collaboration with I. Gaiur and D. Van Straten and with V. Buchstaber and I. Gaiur) about interesting properties of multiplication generalized Bessel kernels, which include well-known Clausen and Sonin-Gegenbauer formulae of XIX century, special examples of Kontsevich discriminant loci polynomials, raised as addition laws for special two-valued formal groups (Buchshtaber-Novikov-Veselov) and period functions solving some Picard-Fuchs equations similar to Calabi–Yau and related to Landau–Ginzburg– like models.
Je ferai un bref état de la recherche (et de mes intérêts personnels) dans le domaine des tenseurs aléatoires et de leurs applications en gravité quantique.
Cette présentation portera sur des chaînes de spins, exemples les plus simples de systèmes intégrables quantiques, dont la résolution utilise les emblématiques "équations de Bethe".
On rappellera les avantages d'une formulation fonctionnelle (relations QQ) de ces équations, et on verra sur deux exemples de chaînes de spins rationelles comment utiliser ces relations fonctionnelles pour construire des expressions explicites des opérateurs T et Q (opérateurs qui interviennent dans la résolution de ces systèmes)