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Homological Invariants of Deformations of Groups and Algebras

Description du projet

Étendre les méthodes d’algèbre homologique à l’étude des systèmes dynamiques

L’algèbre homologique étudie les foncteurs homologiques et les structures algébriques complexes qu’ils impliquent. Elle extrait les informations contenues dans les complexes de chaînes et les présente sous la forme d’invariants homologiques d’anneaux, de modules, d’espaces topologiques et autres objets mathématiques. Financé par le programme Actions Marie Skłodowska-Curie, le projet HIDRA se propose de faire progresser les méthodes d’algèbre homologique en utilisant de nouvelles techniques basées sur les catégories triangulées, la théorie de l’homotopie et la théorie de l’index. Il mettra l’accent sur la conjecture de Baum-Connes, qui sera liée au calcul des invariants K-théoriques et homologiques pour les systèmes dynamiques notables. Les résultats du projet auront un impact important sur les mathématiques pures, la physique du solide et la théorie de l’information quantique.

Objectif

"The pervasive role of algebraic topology in mathematics is proof of the powerful effects that homological invariants produce in the development of the discipline. Extending these techniques beyond the category of topological spaces, in order to include ""quantized"" systems arising from dynamical systems and (quantum) groups, is going to be extremely useful to make fast progress in these fields. The framework of operator algebras and noncommutative geometry is extremely well-suited for these developments and has already been applied with some success. The goal of this proposal is to further develop these homological techniques by supporting them with novel methods based on triangulated categories, homotopy theory, and index theory. The research problems tackled in this Action are deeply related to important topics which attracted a great deal of interest in the mathematical community. For example, we study the celebrated Baum-Connes conjecture (for both groupoids and quantum groups) through a relatively unexplored perspective and relate it to the computation of K-theoretic and homological invariants for notable dynamical systems (e.g. Smale's Axiom A diffeomorphisms). This research will provide mathematicians with both conceptually new approaches and powerful computational tools. Some of these results are relevant not only for pure mathematics, but also for solid-state physics and quantum information theory. This Action will take us one step closer to the solution of significant problems or the formulation of more and more refined research questions. This fellowship will allow V. Proietti to work under the supervision of M. Yamashita (a world-class expert on quantum groups) at the University of Oslo (a leading institution in operator algebras). It will expand the fellow's technical expertise and integrate it with essential management, administrative, and dissemination skills which will help V. Proietti reach a position of professional maturity."

Coordinateur

UNIVERSITETET I OSLO
Contribution nette de l'UE
€ 210 911,04
Adresse
PROBLEMVEIEN 5-7
0313 Oslo
Norvège

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Région
Norge Oslo og Viken Oslo
Type d’activité
Higher or Secondary Education Establishments
Liens
Coût total
Aucune donnée