Singularity theory

Summary

Singularity theory studies the points where a variety or a map fails to be smooth, and classifies them up to local changes of coordinates. Its objects are germs: the behaviour of a function, a map or a variety near one point. Main tools are invariants such as the Milnor number, and resolutions by blowing up. It connects algebraic geometry with differential topology, through Morse functions and catastrophe theory.