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。处理动力系统时最困难的问题之一是如何测量它们的复杂性。在实践中，要处理的信息太多，人们需要使用更简单、更容易计算的不变量，不变量可以洞察周围系统的复杂程度。在代数动力系统领域，即由代数对象(簇)和映射(态射)构造的系统，其中映射被重复地应用于变化中的点，川口和西尔弗曼观察到，一种被称为高度的数论概念可以洞察这类系统的复杂性。他们通过将他们的复杂性算术概念与复杂动力学中使用的更经典的不变量(称为动态度)联系起来，推测地使这一点变得精确。在过去五年中，这已成为算术动力学的一个主要研究领域，而这一建议的大部分内容都建立在他们的框架之上。这项建议涉及显著扩大川口和西尔弗曼的工作范围。在与Dragos Ghioca和Matthew Satriano的联合工作中，我开创了对动力序列的研究，动力序列是从代数动力系统自然获得的序列。从某种意义上说，这种结构给出了动力系统的“快照”，而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，完全了解与给定动力系统相关的所有这样的序列将足以完全重建它。这个广泛的框架考虑了来自数论、代数组合学和微分方程式理论的许多经典结果。我的研究计划的长期目标是回答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驱动的。这些问题包括关于复杂性的问题，关于可判断性的问题，以及关于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，其中一些技术是我在过去六年中帮助开发的，其他技术是由其他研究相关问题的研究人员开发的。这些问题的答案将对许多其他数学学科产生重要的影响，因为它们将在一定程度上为处理许多自然产生的数学问题提供一个算法框架。此外，该提案涉及本科生、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必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.