Abstract

The homotopy method is a very powerful and universal method to prove that a germ (of a function, vector field, map, differential form, or any other object) is $k$-determined, i.e. no terms in its formal series of order bigger than $k$ can change its equivalence class (any natural equivalence relation and any natural category of smoothness is OK). If this is not so then certain modifications of the homotopy method allow to distinguish significant terms of degree bigger than $k$. The objective of the mini-course is as follows: the listeners will be able to use the homotopy method in any local classification problem. In order to achieve that, a number of examples will be detailly considered.