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Automata, Dynamics and Actions
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 a...
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.
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Duración del proyecto: 61 meses
Fecha Inicio: 2023-08-28
Fecha Fin: 2028-09-30
Fecha Fin: 2028-09-30
Línea de financiación: concedida
El organismo HORIZON EUROPE notifico la concesión del proyecto el día 2023-08-28
Línea de financiación objetivo
El proyecto se financió a través de la siguiente ayuda:
ERC-2022-ADG:
ERC ADVANCED GRANTS
Cerrada
hace 2 años
Presupuesto
El presupuesto total del proyecto asciende a
2M€
Líder del proyecto
UNIVERSITAT DES SAARLANDES
No se ha especificado una descripción o un objeto social para esta compañía.
Perfil tecnológico
TRL
4-5
Perfil tecnológico
TRL
4-5