Singularity and regularity for Monge-Ampere type equations

Monge-Ampere方程作为一类重要的非线性偏微分方程组，在几何学、物理学、最优输运等领域都有广泛的应用。许多重要的问题和应用都与解的正则性有关，而解的正则性受到奇点的阻碍。这个项目的目的是对奇异集的几何进行分类，并通过使用创新的方法和发展偏微分方程组的尖端技术来建立一般Monge-Ampere型方程的全面正则性理论。预期成果包括解决悬而未决的问题。该项目将大大加强澳大利亚在数学和应用领域的领导地位和专业知识。

The Monge-Ampere equation, as a premier nonlinear partial differential equation, arises in several areas including geometry, physics, and optimal transportation. Many important problems and applications are related to the regularity of solutions, which are obstructed by singularities. This project aims to classify the geometry of the singular sets, and to establish a comprehensive regularity theory for general Monge-Ampere type equations by using innovative approaches and developing cutting-edge technologies in partial differential equations. Expected outcomes include the resolution of outstanding open problems. This project will significantly enhance Australia’s leadership and expertise in a major area of mathematics and applications.