Singularity and regularity for Monge-Ampere type equations. 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 equation ....Singularity and regularity for Monge-Ampere type equations. 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.Read moreRead less
Stability conditions: their topology and applications. This project aims to answer questions about the topology of the space of stability conditions, which has emerged as a central object in a number of different mathematical areas in the past two decades. The proposed work will have important consequences in representation theory, group theory, and algebraic geometry. The project shows that tools from previously unrelated areas, including discontinous differential equations and discrete dynam ....Stability conditions: their topology and applications. This project aims to answer questions about the topology of the space of stability conditions, which has emerged as a central object in a number of different mathematical areas in the past two decades. The proposed work will have important consequences in representation theory, group theory, and algebraic geometry. The project shows that tools from previously unrelated areas, including discontinous differential equations and discrete dynamical systems, are crucial in the theory of stability conditions. Potential benefits include the resolution of outstanding conjectures in mathematics, the initiation of new connections between different areas of mathematics, and the introduction of machine learning techniques into mathematical research.Read moreRead less