  This is a note on my mini-course in the International Workshop on Real and
Complex Singularities held at ICMC-USP (Sao Carlos, Brazil) in July 2012. Here
we introduce a new branch of the Thom polynomial theory for singularities of
holomorphic maps, in which we replace counting singular points by computing
weighted Euler characteristics. The main purpose is to apply this theory to the
study on the vanishing topology of weighted homogeneous map-germs of finite
A-codimension without any corank condition.
