Monge-Ampère方程常常出现在一些几何问题中，如经典的Minkowski问题，仿射几何中的仿射球等。随着椭圆方程理论的完善，与几何和物理有关的退化与奇异Monge-Ampère型方程引起了广泛的关注，吸引了一批优秀数学家的探索和钻研。 .本项目拟在前期工作的基础上开展以下两方面的研究:Toric流形上极值Kahler度量的存在性问题，镜像对称对构造中的分析问题，建立与之相关的理论。主要解决相关退化与奇异Monge-Ampère型方程解的正则性。

The Monge-Ampere equations often appears in some problems of geometry, such as Minkowski's problem, affine hypersphere in affine geometry etc. With the perfection of the elliptic equations, degenerate and singular equations of Monge-Ampere related to geometry and physics have received wide attention and attracted many mathematicians. In present project, based of previous works, we will study the existence of the extremal Kahler metrics in toric manifolds and the analytic aspect of the construction of mirror dual in mirror symmetry. We will focus on the regularity of solutions of the relevant degenerate and singular equations.