Harmonic analysis for elliptic partial differential equations

Partial differential equations (PDEs) provide a mathematical way of modelling nature. Our previous work have found applications in differential geometry, computational mathematics, optimal transport and more recently in geometric optics. They provided new models for optical systems, which direct given light sources onto desired targets. The current project seeks to establish a good quantitative understanding of solutions to some of the most fundamental types of PDEs. It innovates by constructing a new mathematical object that captures symmetrically the contributions to curvature from different directions, a framework commonly used for expressing image processing operators in computer vision. The project further addresses a longstanding open problem about PDEs, whose study is pioneered by Australian mathematicians. By putting together novel ideas from different areas of mathematics, the project seeks to deliver deep results, and ensure that Australia remains at the cutting edg