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

这个项目将应用模型论和范畴逻辑/拓扑理论的方法，使差分代数取得重大进展，从而导致在动力系统理论中的应用。一个离散的动力系统由一个配备有自映射的空间组成，我们称之为“移位”。在实际应用中，移位映射通常是物理系统的相空间的变换，描述系统从一个时刻到下一个时刻的行为。动力学研究与移位映射迭代过程相关的问题，如（前）周期点的存在性，不变测度，吸引集，混沌行为/对初始条件的敏感依赖等，它在物理学，气象学，生物学，数论和其他纯数学领域有许多应用。它是在20世纪80年代通过可视化的分形，如曼德尔布罗特集普及。我们的项目将触及符号动力学，它研究有限类型的子移位定义为空间的无限字在一个有限的字母表省略许多子字，连同左移位，以及代数动力学，其中移位是一个自同态的代数簇，即，由多变量多项式表达式局部定义。动力系统上的可观测函数的代数被赋予了由移位诱导的自同态，因而可以用差分代数的方法来研究。差环和差模自1930年代开始被研究，当时Ritt将它们定义为具有特殊自同态的环和模。模型论在不同领域的研究中取得了极大的成功。存在闭差域上的可定义集的分类是麦金太尔和查兹达基斯-赫鲁绍夫斯基在齐尔伯的拓扑思想下的工作，通过查兹达基斯-赫鲁绍夫斯基和梅德韦杰夫-斯坎伦的工作对代数动力学产生了深远的影响，后者本质上是处理单变量多项式动力学。我们将研究由“斜积”给出的系统的更困难的情况，其中移位是一个和两个变量的多项式映射的组合。我们将通过使用拓扑理论和分类逻辑，通过将我们的数学的宇宙（基本拓扑）从通常的集合的宇宙改变为差集的宇宙/拓扑，即，装备有自我地图的装置。我们认为里特的差代数结构的代数结构（内部）差集。通过拓扑理论的方法，这个看似微不足道的观察很快导致了深刻的和以前未发现的概念和定理。它允许发展同调代数/上同调理论，代数几何，伽罗瓦理论/etale基本群，etale上同调在不同的背景下，使我们能够制定一个不同的模拟著名的韦伊代数，并作出认真的尝试，其证明。这些抽象的发展为动力系统带来了具体的结果：内部homs和内部自同构群的使用解决了符号动力学中缺乏传递作用的问题，并允许有限类型的子移位的伽罗瓦风格的分类和精确的分解结果，以及树表示理论中的新结果。

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.