O-minimality and dynamical systems

长期以来，数理逻辑在划分形式上的可能性和不可能性方面发挥了重要作用。例如，哥德尔的不完备性定理表明，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它自己的一致性。这种类型的结果限制了数学中形式上可知的范围。近年来，模型理论家已经采取了建构主义的方法来使用数学中的逻辑。在“哥德尔障碍”之下工作，他们已经形式化了结构的简单属性，这些结构具有理想的有限性属性，同时，这些属性产生了丰富的可定义集合。其中最成功的这些性质是一个生活在边界的模型理论和解析几何，即o-极小。这门学科已经显示出它的有用性，它通过神经网络提供了对动力系统、混合系统和学习理论基础的关键见解，并帮助解决了真实的代数几何和数论中的主要开放问题。我的研究动机是调查动力系统的解决方案表现出一定的渐近行为。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，至今仍未解决。我利用o-极小性发展了一种新的方法来解决这个问题，解决了一个非常特殊的问题。在接下来的五年里，我的主要研究方向如下：将我的方法推广到Hilbert第16问题，以获得更重要的情况;简化我们对Pfronan几何的理解，使其更适合于应用;和调查有关的理解从动力系统中产生的现象的O-极小的推广。

Mathematical logic has long played a role in delineating between what is formally possible and impossible. For example, Gödel'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. In recent years, model theorists have taken on a constructivist approach to the use of logic inside mathematics. Working below the "Gödel 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 it. The main lines of my research over the next five years are as follows: generalize my approach to Hilbert's 16th problem 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.