Description du projet
De nouvelles frontières mathématiques pour stimuler l’exploration
Dans le domaine de la dynamique complexe et de la théorie des groupes se trouve le projet ADA financé par le CER. Abordant des questions essentielles en matière de dynamique symbolique, de problèmes de décision et de calcul, ADA utilise des automates à états finis pour définir des objets mathématiques auto-similaires. Le projet introduit le concept de groupes à action automatique, faisant le lien entre des notions auparavant disparates. Qu’il s’agisse de s’attaquer au «problème des lacunes» de Milnor ou d’explorer les groupes non spécifiques associés aux automates, ADA ouvre la voie à de nouvelles stratégies. En encodant des cartes rationnelles par le biais d’actions automatiques, le projet se penche sur le problème de la connexité des longues ouvertures, visant à obtenir une description topologique des «tranches» de cartes quadratiques de Milnor. ADA résout des mystères mathématiques et vise à proposer des solutions aux défis algorithmiques, alimentant l’exploration dans ces univers complexes.
Objectif
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.
Champ scientifique
Programme(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Thème(s)
Régime de financement
HORIZON-ERC - HORIZON ERC GrantsInstitution d’accueil
66123 Saarbrucken
Allemagne