An introduction to Singularity Theory

Singularity Theory has originated (at the end of 19‘th century) with the two basic questions:

• how does the graph of a function look locally?
• how does the zero set of a function look locally?

If the first derivative of a function does not vanish at a point then one can change the local coordinates to linearize the functions. Geometrically one ”rectifies“ the graph/the zero set. Accordingly the local geometry/topology/algebra are trivialized. The situation becomes much more involved when the first derivative vanishes. I will consider several simple (though non-trivial) examples, showing how the algebra/geometry/topology are involved.

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