Ricci-flat metrics on toric non-compact Calabi-Yau manifolds

The talk will be dedicated to the methods of constructing Kдhler Ricci-flat metrics on certain non-compact Calabi-Yau manifolds. We will be mainly dealing with the total space of the canonical bundle over the blow-up of $\mathbb C \mathbb P^2$ at one point. We will also discuss some applications of these metrics in theoretical physics (in the context of supersymmetric solutions of the equations of motion of supergravity theories).