Descripción del proyecto
Nuevas fronteras matemáticas para impulsar la exploración
El proyecto ADA, financiado por el CEI, se sitúa en el ámbito de la dinámica compleja y la teoría de grupos. Al abordar cuestiones fundamentales de dinámica simbólica, problemas de decisión y cálculo, el equipo de ADA utiliza autómatas de estado finito para definir objetos matemáticos autosimilares. Introduce el concepto de grupos de actuación automática, que tiende puentes entre nociones hasta ahora dispares. Desde abordar el «problema de la brecha» de Milnor hasta explorar grupos no sólidos asociados a los autómatas, el equipo de ADA es pionero en estrategias novedosas. Al codificar mapas racionales mediante acciones automáticas, en el proyecto se profundiza en el problema de la conectividad entre espacios abiertos, con el objetivo de lograr una descripción topológica de las «rebanadas» de los mapas cuadráticos de Milnor. El equipo de ADA resuelve misterios matemáticos y pretende proponer soluciones a retos algorítmicos, impulsando la exploración en estos universos intrincados.
Objetivo
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.
Ámbito científico
Palabras clave
Programa(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Régimen de financiación
HORIZON-ERC - HORIZON ERC GrantsInstitución de acogida
66123 Saarbrucken
Alemania