  These notes have two parts. The first is a study of Nekrasov's deformed
partition functions $Z(\ve_1,\ve_2,\vec{a};\q,\vec{\tau})$ of N=2 SUSY
Yang-Mills theories, which are generating functions of the integration in the
equivariant cohomology over the moduli spaces of instantons on $\mathbb R^4$.
The second is review of geometry of the Seiberg-Witten curves and the geometric
engineering of the gauge theory, which are physical backgrounds of Nekrasov's
partition functions.
  The first part is continuation of math.AG/0306198, where we identified the
Seiberg-Witten prepotential with $Z(0,0,\vec{a};\q,0)$.
  We put higher Casimir operators to the partition function and clarify their
relation to the Seiberg-Witten $u$-plane. We also determine the coefficients of
$\ve_1\ve_2$ and $(\ve_1^2+\ve_2^2)/3$ (the genus 1 part) of the partition
function, which coincide with two measure factors $A$, $B$ appeared in the
$u$-plane integral.
  The proof is based on the blowup equation which we derived in the previous
paper.
