Conférence de Boris Adamczewski, CNRE et Université de Lyon
Finite automata form a class of very basic Turing machines. In number theory, they can be used to define in a natural way sequences and sets which are said to be 'automatic'. One of the main interest of these automatic structures is that they enjoy some strong regularity without being trivial at all. They can be thus though of as lying somewhere between order and chaos, though in many aspects they appear as essentially regular.
This special feature of automatic structures leads to various applications of automata theory to number theory. As part of my Aisenstadt chair, I will give a series of lectures describing some links between these automatic structures and some classical number theoretical problems. Such problems include the representation of integers and real numbers in an integer base, Diophantine equations and decidability, the study of arithmetic differential equations, transcendence and algebraic independence. Researches in this area are currently funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under the grant Agreement No 648132.