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

差分代数的模型理论和拓扑理论观点及其在动力学中的应用

• 批准号: EP/V028812/1
• 负责人: Ivan Tomasic
• 金额: $ 60.35万
• 依托单位: Queen Mary University of London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FV028812%2F1
• 关键词: Model; theoretic; topos; view; difference

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.

该项目将应用模型论和范畴逻辑/拓扑论的方法，在差分代数方面取得重大进展，从而在动力系统理论中得到应用。离散动力系统由配备有自映射的空间组成，我们称之为“移位”。在现实世界的应用中，移位图通常是物理系统相空间的变换，它描述了系统从一个时刻到下一个时刻的行为。动力学研究与迭代移位映射有关的问题，如(准)周期点的存在性、不变度量、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、生物学以及数论和其他纯数学领域有着广泛的应用。它是在20世纪80年代通过对诸如Mandelbrot集的分形的可视化而流行起来的。我们的项目将涉及符号动力学，它研究定义为有限字母表中省略有限多个子词的无限单词空间的有限类型子移位，以及左移，以及代数动力学，其中移位是代数种类的自同态，即由多元多项式表达式局部定义。动力系统上可观测函数的代数被赋予了由移位引起的自同态，因此可以用差分代数的方法来研究它们。差环和模的研究始于20世纪30年代，当时Ritt将其定义为具有特殊自同态的环和模。模型理论在不同领域的研究中取得了极大的成功。在Zilber三分法的精神下，Macintyre和Chatzidakis-Hrushovski的工作产生了存在闭差域上的可定义集的分类，这一分类通过Chatzidakis-Hrushovski和Medvedev-Scanlon的工作在代数动力学中产生了深刻的影响，后者基本上处理一元多项式动力学。我们将研究一个更为困难的系统，它是一个变量和两个变量的多项式映射的组合。我们将通过使用Topos理论和范畴逻辑来革命差分代数，将我们数学的宇宙(基本Topos)从惯常的集合宇宙改变为差集合的宇宙/Topos，即配备自映射的集合。我们把Ritt的差分代数结构看作是差集中的(内部)代数结构。通过拓朴理论的方法，这种看似微不足道的观察很快就带来了以前未被发现的深刻概念和定理。它允许发展同调代数/上同调理论、代数几何、Galois理论/等基本群，以及在不同背景下的上同调，使我们能够对著名的Weil猜想进行不同的模拟，并对其进行认真的证明。这些抽象的发展为动力系统产生了具体的结果：内HOMS和内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，并允许有限型子移位的伽罗瓦式分类和精确分解结果以及树表示理论中的新结果。

项目成果

Difference Galois theory and dynamics

伽罗瓦理论与动力学的区别

• DOI: 10.1016/j.aim.2022.108328
• 发表时间: 2022
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Tomašic I