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.

数理逻辑长期以来一直在区分形式上可能和不可能的方面发挥着作用。例如，哥德尔的不完备性定理表明，没有足够复杂的一致形式系统可以证明数学中的简单定理可以证明其自身的一致性。这种类型的结果限制了数学中形式上已知的范围。在过去的三十年里，模型理论家采取了建构主义的方法来在数学中使用逻辑。在“哥德尔势垒”下方工作，他们形式化了具有理想有限性的结构的简单属性，同时产生了丰富的可定义集合。 这些性质中最成功的一个位于模型理论和解析几何的边缘，即 o 极小性。该学科已经通过神经网络提供对动力系统、混合系统和学习理论基础的关键见解而显示出其实用性，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。我对希尔伯特第 16 个问题特别感兴趣，它是德国数学家大卫·希尔伯特在 1900 年提出的著名的 23 个问题之一；至今仍未解决。我使用 o-极小性开发了一种解决此问题的新方法，解决了 Roussarie 猜想的一个非常特殊的情况（希尔伯特第 16 个问题的本地化陈述）。 在过去的五年里，我已经完成了推广我的方法以获得更多关于罗萨里猜想的重要案例的第一步。 这部分是在 Tobias Kaiser（德国帕绍）和我的硕士生 Zeinab Galal 的合作下完成的。 未来五年我的研究主线如下：对罗萨里猜想进行下一步研究，以获得更多有意义的事例；简化我们对普法夫几何的理解，使其更适合应用；以及与理解动力系统产生的现象相关的 o-极小性概括的研究。