O-minimal structures and dynamical systems

长期以来，数学逻辑一直在描述形式上可能与不可能之间的关系。例如，哥德尔不完备定理表明，没有一个复杂到足以证明数学中简单定理的一致形式系统可以证明其自身的一致性。这种结果限制了数学中形式上可知的范围。在过去的三十年里，模型理论家采取了一种建构主义的方法来使用数学中的逻辑。在“Go¨del障碍”下工作，他们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集合。******在这些性质中，最成功的是一个存在于模型理论和解析几何的边界上的性质，即极小性。这门学科已经通过提供对动力系统、混合系统和通过神经网络学习理论的基础的关键见解，并帮助解决了实际代数几何和数论中的主要开放问题，显示了它的有用性。******我研究的动机是研究动力系统，其解表现出某些渐近行为。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年提出的著名的23个问题之一；这个问题至今仍未解决。我开发了一种新的方法来解决这个问题，使用o-minimality，解决了Roussarie猜想的一个非常特殊的情况（Hilbert's 16问题的一个局部陈述）。在过去的5年里，我已经完成了推广我的方法的第一步，以获得更多的Roussarie猜想的重要案例。这项研究部分是由Tobias Kaiser（德国帕绍）和我的硕士学生Zeinab Galal合作完成的。******未来五年我的研究主线如下：在我对Roussarie猜想的研究中进行下一步，以获得更多有意义的案例；简化我们对普法几何的理解，使其更适合应用；以及与理解动力系统产生的现象有关的o-极小性的推广研究。*****

Mathematical logic has long played a role in delineating between what is formally possible and impossible. For example, Go¨del'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 "Go¨del 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.*****