gromov witten theory - Does any one understand the details of M Kazarian's work in enumerative geometry of $mathbb{C}mathbb{P}^2$ ?

I wanted to know if anyone understood the details of the paper

"Multisingularities, cobordisms, and enumerative geometry" available at the site

http://www.mi.ras.ru/~kazarian/.

In particular does any one follow how the author actually got all the expressions
for $S_{A_k}$ on page 47. I think I follow the details of how he got $S_{A_1}$.
But how does he get $S_{A_1^2}$ for instance?