Descrizione del progetto
Nuove frontiere matematiche per dare impulso alla ricerca
Nel regno della dinamica complessa e della teoria dei gruppi si colloca il progetto ADA, finanziato dal CER. ADA affronta questioni cruciali nell’ambito della dinamica simbolica, dei problemi decisionali e del calcolo, usando automi a stati finiti per definire oggetti matematici autosomiglianti. Inoltre, introduce il concetto di gruppi che agiscono automaticamente, collegando nozioni precedentemente eterogenee. Affrontando questioni come il «problema della lacuna» di Milnor e l’esplorazione di gruppi non sofici associati agli automi, ADA si fa pioniere di nuove strategie. Codificando le mappe razionali attraverso azioni automatiche, il progetto si addentra nel problema della connessione lunga-aperta, mirando a una descrizione topologica delle «fette» di mappe quadratiche di Milnor. ADA risolve misteri matematici e mira a proporre soluzioni alle sfide algoritmiche, stimolando l’esplorazione di questi intricati universi.
Obiettivo
This project lies at the nexus of complex and symbolic dynamics, group theory, decision problems and computation. It aims to solve major problems in each of these fields by means of automatic actions and relations.
Finite state automata, pervasive in theoretical computer science, will serve to define self-similar mathematical objects, and produce efficient algorithms to manipulate them. — I will explore a novel notion of automatically acting group, encompassing the previously unrelated notions of automatic groups, automata groups and substitutive shifts.
Geometric group theory propounds the vision of groups as geometric objects. A basic notion is volume growth, and Milnor's still open “gap problem” asks for its possible range. — In this proposal, I will give candidates of groups with very slow superpolynomial growth, defined by their automatic action on dynamical systems, and a proof strategy.
A celebrated open problem by Gromov asks whether all groups are “sofic”. This property has too many valuable consequences to always be true, yet there is no known non-example! — I will present a strategy of producing non-sofic groups closely associated to automata.
Rational maps on the Riemann sphere provide a rich supply of dynamical systems. A fundamental goal is to give a combinatorial description of the dynamics across families of maps, constructing models of parameter space. — I will encode the maps via automatic actions, and study relations between automata to produce such models. I aim to achieve a full topological description (including the long-open connectedness problem) of Milnor's “slices” of quadratic maps.
This project will tackle these fundamental questions from group theory and dynamics, and develop presently unexplored interactions between them, through a unified use of automata. It will prove decidability of certain algorithmic problems such as Dehn's and Tarski's, and construct efficient tools to further our exploration of these mathematical universes.
Campo scientifico
Parole chiave
Programma(i)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Argomento(i)
Meccanismo di finanziamento
HORIZON-ERC - HORIZON ERC GrantsIstituzione ospitante
66123 Saarbrucken
Germania