A priori estimates for Monge-Amp\`ere equation and applications.

In this talk we will discuss the a priori estimates for some Monge-Amp\`ere type equation and their applications. We begin with convergence rate estimates for the Monge-Amp\`ere equation. Following this we study a singular Monge-Amp\`ere equation, whose solvability impies the existence of affine hyperspheres asymptotic to some cones as $x\to\infty$. Finally, we disscuss the $L_p$ dual Minkowski problem. We will use parabolic flow and variational method to study the exsitence, regularity and uniqueness of the solutions to the problem.