Objectif During his construction of a solid mathematical theory behind - the at that time completely new - quantum mechanics, von Neumann introduced his eponymous algebras to describe observable quantities. These “von Neumann algebras” became a basic tool in various other branches of mathematics, including Lie theory (the theory of continuous symmetries), non-commutative geometry (a “quantum” version of classical differential geometry), and, surprisingly, the theory of knots, for which V. Jones received a Fields Medal. Strangely enough, although the theory of von Neumann algebras is quite pervasive in mathematics and mathematical physics, their actual construction and classification remains largely shrouded in mystery (despite deep work on classification by A. Connes, also getting him a Fields Medal). Particularly unsatisfactory is that the types of von Neumann algebras that are most relevant to quantum mechanics, so-called “type III”-algebras, are very rare. With this Marie-Curie fellowship, I pick up the two challenges of construction and classification, especially focussing on Connes' famous rigidity conjecture for lattices in Lie groups as well as type III von Neumann algebras, using two entirely new approaches. The first is the use of finite-dimensional approximations, that I used previously in a different context (studying the Haagerup property and Lp-Fourier multipliers). The second new approach is based on the theory of quantum groups. Utrecht University (host institution) is the unique place in Europe housing both experts in non-commutative analysis and Lie theory, and thereby provides exactly the necessary (complementary) expertise that is necessary to attack these deep and profound problems.The results will have a lasting impact on and connect further the theories of non-commutative geometry, operator algebras, Lie theory, quantum group theory and partly quantum physics. Champ scientifique natural sciencesphysical sciencesquantum physicsnatural sciencesmathematicsapplied mathematicsmathematical physicsnatural sciencesmathematicspure mathematicsmathematical analysisfunctional analysisoperator algebranatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsalgebraalgebraic geometry Mots‑clés Quantum groups Von Neumann algebras Operator spaces Approximation properties Programme(s) H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions Main Programme H2020-EU.1.3.2. - Nurturing excellence by means of cross-border and cross-sector mobility Thème(s) MSCA-IF-2015-EF - Marie Skłodowska-Curie Individual Fellowships (IF-EF) Appel à propositions H2020-MSCA-IF-2015 Voir d’autres projets de cet appel Régime de financement MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF) Coordinateur UNIVERSITEIT UTRECHT Contribution nette de l'UE € 165 598,80 Adresse HEIDELBERGLAAN 8 3584 CS Utrecht Pays-Bas Voir sur la carte Région West-Nederland Utrecht Utrecht Type d’activité Higher or Secondary Education Establishments Liens Contacter l’organisation Opens in new window Site web Opens in new window Participation aux programmes de R&I de l'UE Opens in new window Réseau de collaboration HORIZON Opens in new window Coût total € 165 598,80
