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Singularity theory

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Let be the euclidian space . Let be a subset of . If is a submanifold at there is a local canonical form for at there is a diffeomorphism from an open neighborhood of into an open set of that takes into the intersection of with a linear subspace of . The set looks like a linear subspace in a neighborhood of .Let be a smooth vector field on . If does not vanish at a point , there is a local canonical form for at there is a diffeomorphism from an open neighborhood of into an open set of that takes the restriction of to into a constant vector field on . The vector field looks like a constant vector field in a neighbourhood of .Singularity theory looks at what happens at the points where is a not a submanifold of or vanishes.At these points things get more complicated. In order to say something we will need some additional hypothesis. We will ask for instance that is defined by polynomial equations and has polynomial coefficients. We can replace the field by an arbitrary field . The theory becomes much simpler we we assume the field is algebraically closed. The first singularities to be studied were the singularities of plane curves. Its natural generalization, the study of isolated singularities of complex analýtic hypersurface singularities is the core of Singularity theory. There is no good concise description of Singularity Theory. We can enumerate some of the objects it studies: algebraic sets, analytic sets, algebraic maps, analytic maps, vector fields, differential forms, Lagrangian varieties... We can enumerate some of the questions that are usually asked: do we have topological [smooth, analytic] canonical forms? can we resolve [reduce] its singularities? what are its deformations? what are the objects that have no deformations [only a finite number of non-isomorphic deformations]? On the other hand Singularity theory does not study an isolated singularity of an holomorphic function. There is no geometric or topological flavor in this problem.

Wikipedia Inglés http://en.wikipedia.org/wiki/Singularity_theory

Wikipedia Español http://es.wikipedia.org/wiki/Singularity_theory