Archimedean and Non-Archimedean Higher Order Stratifications

奇点是一个几何物体“不光滑”的点，就像锥体的尖端或一张纸的折痕。奇点理论的目标是描述当某一类方程给出几何对象时，会产生什么样的奇点。当方程组模拟一个来自现实世界的系统（如天气、人口密度、机械臂的位置）时，奇点通常对应于一个具有突然变化和/或不可预测行为的状态。当给定一个几何物体时，人们通常想要对它的所有奇点进行分类，并描述奇点在该物体上的分布。其中一个数学工具是“分层”。有不同种类的分层，产生不同精度的奇点分类。然而，即使是最著名的分层，对于某些预期的应用也还不够好。该项目的目标是利用以下基于逻辑的新方法开发更精确的分层。为了分析一个几何物体某一点上的奇点，我们考虑这个几何物体在该点的一个小邻域内：邻域越小越好，因为在更大的邻域中，我们会看到更多与奇点本身无关的东西。通常，考虑点的“无限小”邻域是没有意义的，因为它只由点本身组成。然而，利用逻辑的方法，可以引入无限小的数，这样，无限小的邻域也变得有意义了；特别地，它们在描述奇点时变得非常方便。最精确的经典分层概念（不使用无限小的邻域）的定义是极其复杂和技术性的。在过去的几年里，通过使用逻辑方法，我发展了几个新的分层概念，这些概念在精度上与最好的经典概念相似，但更容易定义和更容易使用。这个项目的计划是进一步改善这些新的分层。

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.