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Homological Invariants of Deformations of Groups and Algebras

Descrizione del progetto

Estendere i metodi dell’algebra omologica allo studio dei sistemi dinamici

L’algebra omologica studia i funtori omologia e le intricate strutture algebriche che essi comportano. Essa estrae le informazioni contenute nei complessi a catena e le presenta sotto forma di invarianti omologici di anelli, moduli, spazi topologici e altri oggetti matematici. Il progetto HIDRA, finanziato dal programma di azioni Marie Skłodowska-Curie, intende promuovere ulteriormente i metodi dell’algebra omologica utilizzando nuove tecniche basate sulle categorie triangolate, sulla teoria dell’omotopia e sulla teoria degli indici. Ci si concentrerà sulla congettura di Baum-Connes, che sarà collegata al calcolo degli invarianti K-teorici e omologici per i sistemi dinamici degni di nota. I risultati del progetto avranno un impatto importante sulla matematica pura, sulla fisica dello stato solido e sulla teoria dell’informazione quantistica.

Obiettivo

"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."

Meccanismo di finanziamento

HORIZON-AG-UN - HORIZON Unit Grant

Coordinatore

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

Mostra sulla mappa

Regione
Norge Oslo og Viken Oslo
Tipo di attività
Higher or Secondary Education Establishments
Collegamenti
Costo totale
Nessun dato