Heights, Dynamics, and Decidability

动力系统是重要的数学研究对象，由于其在建模随时间变化的复杂系统中的实用性，现在在许多物理科学和社会科学中无处不在。 在处理动力系统时，最困难的问题之一是如何度量它们的复杂性。 在实践中，有太多的信息需要处理，人们需要使用更简单，更容易计算的不变量来了解环境系统的复杂程度。 在领域的代数动力系统，即系统是由代数对象（品种）和地图（态射）和地图是重复适用于点的品种，这是观察到的川口和西尔弗曼的数论概念，所谓的高度，让洞察复杂性的系统。他们通过将复杂性的算术概念与复杂动力学中使用的更经典的不变量（称为动力学度）联系起来，从而精确地实现了这一点。 这已经成为一个主要的研究领域内的算术动力学在过去的五年中，这一建议的大部分建立在他们的框架。这一建议涉及大大扩大川口和西尔弗曼的工作范围。 在联合工作与Dragos吉奥卡和马修萨特里亚诺，我开始研究动态序列，这是序列，自然获得的代数动力系统。 从某种意义上说，这种构造给出了动力系统的“快照”，这些序列编码了有关环境动力系统复杂性的信息。 从理论上讲，拥有与给定动力系统相关的所有此类序列的完整知识将足以完全重建它。这个广泛的框架认为许多经典的结果从数论，代数组合，并从微分方程理论。我的研究计划的长期目标是回答在这个框架内出现的关键问题，这些问题是由许多不同数学学科的基本问题所激发的。这些问题包括关于复杂性的问题，关于可判定性的问题，以及丢番图近似和逻辑的问题。我将通过应用最新的技术来回答这些问题，其中一些是我在过去六年中帮助开发的，另一些是由其他研究相关问题的研究人员开发的。 这些问题的答案将在许多其他数学学科中产生重要的影响，因为它们将部分地为处理许多自然产生的数学问题提供算法框架。 此外，该提案还涉及本科生，研究生和博士后研究人员的培训，并将为学员提供重要的数学专业知识以及在STEM相关行业和学术界工作所需的编程和写作技能。

Dynamical systems are important mathematical objects of study, which are now ubiquitous within many of the physical sciences and social sciences due to their utility in modelling complex systems that vary over time.  One of the most difficult problems when dealing with dynamical systems is how to measure their complexity.  In practice, there is too much information to work with and one needs to work with simpler, easier to compute, invariants which give insight into how complex the ambient system is.  In the realm of Algebraic dynamical systems, that is, systems which are constructed from algebraic objects (varieties) and maps (morphisms) and where a map is repeatedly applied to points in the variety, it was observed by Kawaguchi and Silverman that a number theoretic notion, called the height, gives insight into the complexity of such systems. They have conjecturally made this precise by linking their arithmetic notion of complexity to a more classical invariant used within complex dynamics, called the dynamical degree.  This has gone on to become a major area of research within arithmetic dynamics within the last five years and much of this proposal builds upon their framework. This proposal deals with significantly broadening the scope of Kawaguchi and Silverman's work.  In joint work with Dragos Ghioca and Matthew Satriano, I initiated the study of dynamical sequences, which are sequences that are naturally obtained from algebraic dynamical systems.  In a sense, this construction is giving a "snapshot" of the dynamical system and these sequences encode information about the complexity of the ambient dynamical system.  In theory, having complete knowledge of all such sequences associated to a given dynamical system would be sufficient to reconstruct it entirely. This broad framework considers many classical results from number theory, algebraic combinatorics, and from the theory of differential equations. The long-term goal of my research program is to answer key questions that arise within this framework, which are motivated by fundamental problems across many different mathematical disciplines. These include questions about complexity, questions about decidability, and questions in Diophantine approximation and logic. I will answer these questions by applying recent techniques, some of which I have helped develop over the past six years, and others that have been developed by other researchers working on related problems. Answers to these questions will have important ramifications in many other mathematical disciplines, as they will in part provide an algorithmic framework for dealing with many naturally arising mathematical problems.  In addition, the proposal involves the training of undergraduate, graduate, and postdoctoral researchers, and will give trainees important mathematical expertise and both the programming and writing skills that are necessary for working in both STEM-related industries and in academia.