Model theoretic and topos theoretic view of difference algebra and applications to dynamics

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This project will apply methods of model theory and categorical logic/topos theory to make significant advances in difference algebra, which will consequently result in applications in the theory of dynamical systems. A discrete dynamical system consists of a space equipped with a self-map we call the`shift'. In real-world applications, the shift map is usually a transformation of the phase space of a physical system that describes the behaviour of the system from one moment to the next. Dynamics studies questions related to the process of iterating the shift map, such as the existence of (pre)periodic points, invariant measures, attracting sets, chaotic behavious/sensitive dependence on initial conditions, etc. It has numerous applications in physics, meteorology, biology, but also in number theory and other areas of pure mathematics. It was popularised in the 1980s through visualisations of fractals such as the Mandelbrot set.Our project will touch upon symbolic dynamics, which studies subshifts of finite type defined as spaces of infinite words in a finite alphabet omitting finitely many subwords, together with the left shift, as well as algebraic dynamics, where the shift is an endomorphism of an algebraic variety, i.e., locally defined by multivariate polynomial expressions. The algebras of observable functions on dynamical systems are endowed with the endomorphism induced by the shift and hence they can be studied by methods of difference algebra. Difference rings and modules have been studied since the 1930s, when Ritt defined them as rings and modules endowed with distinguished endomorphisms. Model theory has been extremely successful in the study of difference fields. A classification of definable sets over existentially closed difference fields that emerged from the work of Macintyre and Chatzidakis-Hrushovski in the spirit of Zilber's trichotomy has had a deep impact in algebraic dynamics through work of Chatzidakis-Hrushovski and Medvedev-Scanlon, where the latter essentially treats the univariate polynomial dynamics. We will study the much more difficult case of systems given by `skew-products', where the shift is a combination of polynomial maps in one and two variables.We will revolutionise difference algebra through the use of topos theory and categorical logic by changing the universe (base topos) for our mathematics from the customary universe of Sets to the universe/topos of difference sets, i.e., sets equipped with a self-map. We view Ritt's difference algebraic structures as algebraic structures (internal) in difference sets. Through the methods of topos theory, this seemingly trivial observation quickly leads to deep and previously undiscovered concepts and theorems. It allows the development of homological algebra/cohomology theory, algebraic geometry, Galois theory/etale fundamental group, etale cohomology in the difference context, allowing us to formulate a difference analogue of the celebrated Weil conjectures and make a serious attempt at its proof. These abstract developments yield concrete consequences for dynamical systems: the use of internal homs and internal automorphism groups resolves the issues on the lack of transitive actions in symbolic dynamics, and allows a Galois-style classification and precise decomposition results for subshifts of finite type and new results in the theory of arboreal representations.