Cerf theory

In mathematics, at the junction of singularity theory and differential topology, Cerf theory is the study of families of smooth real-valued functions

${\displaystyle f\colon M\to \mathbb {R} }$

on a smooth manifold ${\displaystyle M}$, their generic singularities and the topology of the subspaces these singularities define, as subspaces of the function space. The theory is named after Jean Cerf, who initiated it in the late 1960s.