Structural Stability and Morphogenesis of Differentiable Maps

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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.