Archimedean and Non-Archimedean Higher Order Stratifications

阿基米德和非阿基米德高阶分层

• 批准号: 426488848
• 负责人: Professor Dr. Immanuel Halupczok
• 金额: --
• 依托单位: Lehrstuhl für Algebra und Zahlentheorie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/426488848?language=en
• 关键词: Archimedean; Non; Higher; Order; Stratifications

A singularity is a point where a geometric object is "not smooth", like the tip of a cone or a fold in a sheet of paper. The goal of singularity theory is to describe what kind of singularities can arise when the geometric object is given by certain kinds of equations. When the equations model a system from the real world (like the weather, population densities, positioning of a robot arm), a singularity typically corresponds to a state with sudden changes and/or unpredictable behavior.When a geometric object is given, one typically would like to classify all its singularities, and also to describe how the singularities are distributed on the object. One mathematical tool for this are "stratifications". There are different kinds of stratifications, which yield classifications of the singularities of different precision. However, even the best known stratifications are not yet good enough for certain intended applications. The goal of this project is to develop more precise stratifications using the following new approach based on logic.To analyze a singularity at a certain point of a geometric object, one considers the geometric object in a tiny neighborhood of that point: the smaller the neighborhood, the better, since on larger neighborhoods, one sees more things that are not relevant for the singularity itself. Usually, considering an "infinitely small" neighbourhood of the point would not make sense, since it would only consist of the point itself. However, using methods from logic, one can introduce infinitely small numbers, so that afterwards, also infinitely small neighbourhoods become meaningful; in particular, they become very handy to describe singularities.The most precise classical notions of stratifications (which do not use infinitely small neighbourhoods) are extremely complicated and technical to define. In the last years, by using the approach via logic, I developed several new notions of stratifications which are similar in precision as the best classical ones, but much simpler to define and much easier to work with. The plan for this project is to further improve those new stratifications.

奇点是几何对象“不光滑”的点，例如圆锥体的尖端或一张纸的折叠处。奇点理论的目标是描述当几何对象由某些类型的方程给定时会出现什么样的奇点。当方程对现实世界的系统（如天气、人口密度、机器人手臂的位置）进行建模时，奇点通常对应于突然变化和/或不可预测行为的状态。当给定一个几何对象时，人们通常希望对其所有奇点进行分类，并描述奇点在对象上的分布方式。一种数学工具是“分层”。存在不同种类的分层，从而产生不同精度的奇点的分类。然而，即使是最知名的分层对于某些预期应用来说还不够好。该项目的目标是使用以下基于逻辑的新方法开发更精确的分层。为了分析几何对象某一点的奇点，我们需要考虑该点的微小邻域中的几何对象：邻域越小越好，因为在更大的邻域中，人们会看到更多与奇点本身无关的事物。通常，考虑点的“无限小”邻域是没有意义的，因为它仅由点本身组成。然而，使用逻辑方法，我们可以引入无限小的数字，这样之后，无限小的邻域也变得有意义；特别是，它们对于描述奇点变得非常方便。最精确的经典分层概念（不使用无限小的邻域）的定义极其复杂且技术性强。在过去的几年里，通过使用逻辑方法，我开发了几种新的分层概念，这些概念在精度上与最好的经典概念相似，但定义更简单，也更容易使用。该项目的计划是进一步改善这些新的分层。