Archimedean and Non-Archimedean Higher Order Stratifications

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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.