O-minimal structures and dynamical systems

O-最小结构和动力系统

• 批准号: RGPIN-2018-06555
• 负责人: Speissegger, Patrick
• 金额: $ 1.46万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712861
• 关键词: minimal; structures; dynamical; systems

Mathematical logic has long played a role in delineating between what is formally possible and impossible. For example, Godel's incompleteness theorem shows that no consistent formal system sophisticated enough to prove simple theorems in mathematics can demonstrate its own consistency. This type of result places a limit on the scope of what is formally knowable in mathematics. Over the last thirty years, model theorists have taken on a constructivist approach to the use of logic inside mathematics. Working below the "Godel barrier'', they have formalized simple properties of structures that have desirable finiteness properties and which, at the same time, give rise to a rich collection of definable sets. Among the most successful of these properties is one that lives on the border of model theory and analytic geometry, namely, o-minimality. This subject has already shown its usefulness by providing key insights into the foundations of dynamical systems, hybrid systems, and learning theory through neuronal networks and has helped settle major open problems in real algebraic geometry and number theory. The motivation for my research is the investigation of dynamical systems whose solutions exhibit certain asymptotic behaviours. I am particularly interested in Hilbert's 16th problem, one of the famous list of 23 problems posed by the German mathematician David Hilbert in 1900; it remains unsolved to this day. I have developed a new approach to this problem using o-minimality, solving a very special case of Roussarie's conjecture (a localized statement of Hilbert's 16th problem). Over the past 5 years, I have completed the first step towards generalizing my approach to obtain more significant cases of Roussarie's conjecture. This was done in parts with collaboration of Tobias Kaiser (Passau, Germany) and my MSc student Zeinab Galal. The main lines of my research over the next five years are as follows: carry out the next steps in my approach to Roussarie's conjecture to obtain more significant cases of it; simplify our understanding of Pfaffian geometry, in order to make it more suitable for applications; and the investigation of generalizations of o-minimality relevant to the understanding of phenomena arising from dynamical systems.

长期以来，数理逻辑在区分形式上可能和不可能之间扮演着重要角色。例如，哥德尔的不完备性定理表明，没有一个协调一致的形式系统能够证明自己的一致性，该系统足够复杂，足以证明数学中的简单定理。这种类型的结果限制了数学中形式上可知的范围。在过去的三十年里，模型理论家已经采取了一种建构主义的方法来使用数学中的逻辑。在“哥德尔屏障”下工作，他们已经形式化了结构的简单性质，这些结构具有理想的有限性质，同时，产生了丰富的可定义集合。 在这些性质中，最成功的是一种生活在模型理论和解析几何之间的性质，即o-极小。这门学科已经通过提供对动力系统、混合系统和通过神经元网络的学习理论的基础的关键见解来展示其实用性，并帮助解决了实代数几何和数论中的主要开放问题。 我研究的动机是研究其解表现出某些渐近行为的动力系统。我对希尔伯特的第16个问题特别感兴趣，这是德国数学家大卫·希尔伯特在1900年提出的23个著名问题之一；它至今仍未解决。我已经开发了一种使用o-最小性来解决这个问题的新方法，解决了Roussarie猜想的一个非常特殊的情况(希尔伯特第16个问题的局部化陈述)。在过去的5年里，我已经完成了推广我的方法的第一步，以获得更多重要的卢萨里猜想案例。这部分是在托拜厄斯·凯泽(德国帕索)和我的硕士学生Zeinab Galal的合作下完成的。 在接下来的五年里，我的研究主线如下：在我处理Roussarie猜想的方法中执行下一步，以获得它的更重要的例子；简化我们对Pfaffian几何的理解，以便使它更适合于应用；以及研究与理解动力系统产生的现象相关的o-极小的推广。