Stable mappings of manifolds (stable mappings of henselian germs of schemes)

Whitney studied the embeddings of (C^\infty) manifolds into R^N. A simple initial idea is: start from a map M-> R^N, and deform it generically. Hopefully one gets an embedding, at least an immersion. This fails totally because of the “stable maps”. They are non-immersions, but are preserved in small deformations. The theory of stable maps was constructed in 50’s-60’s by Thom, Mather and others. The participating groups are infinite-dimensional, and the engine of the theory was vector fields integration. This chained all the results to the real/complex-analytic case. I will discuss the classical case, then report on the new results, extending the theory to the arbitrary field (of any characteristic).