Structural Stability and Morphogenesis of Differentiable Maps

本研究项目的目的是利用一个通用的框架，从一个新的角度来表述流形和映射的不变量。为此，我们考虑了奇点或奇异纤维的每种分类的Vassiliev复合体，并研究了它们的不变量。进一步，我们将具有边界的3-流形稳定映射的奇异纤维完全分类到平面上。然后我们计算了相关Vassiliev复合体的上同群，并发现了新的共不变量。此外，我们将这种研究应用于多变量数据的可视化，并表明这种技术对于阐明给定大数据集的微分拓扑特征非常有用。

The purpose of this research project was to formulate invariants of manifolds and maps from a new viepoint using a universal framework. For this, we considered Vassiliev complex for each classification of singularities or singular fibers and studied their invariants. Furthermore, we completely classified singular fibers of stable maps of 3-manifolds with boundary into the plane. We then calculated the cohomology groups of the associated Vassiliev complex and discovered new cobordism invariants. Moreover, we applied this kind of study to the visualization of multivariate data, and we showed that this kind of techniques can be very useful for clarifying differential topological features of a given set of big data.