Introduction to Singularity Theory

1. An introductory sketch and some motivating examples. Degenerate critical points of functions. Singular (nonsmooth) points of curves.
2. Holomorphic functions of several variables. Weierstrass preparation theorem. Local Rings and germs of functions/sets.
3. Isolated critical points of holomorphic functions. Unfolding and morsication. Finitely determined function germs.
4. Classification of simple singularities. Basic singularity invariants. Plane curve singularities. Decomposition into branches and Puiseux expansion.
5. Time permitting we will concentrate on some of the following topics: a. Blowups and resolution of plane curve singularities; b. Basic topological invariants of plane curve singularities (Milnor fibration); c. Versal deformation and the discriminant.

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