Billiards within confocal quadrics and beyond. This project aims to analyse mathematical billiards within domains bounded by confocal conics. Mathematical billiards have applications in any situation that involves collisions and reflections, and any phenomenon that includes reflections and collisions can be modelled using mathematical billiards. This project aims to revolutionise the analysis of billiards within domains bounded by several confocal conics by exploring the relations of such billia ....Billiards within confocal quadrics and beyond. This project aims to analyse mathematical billiards within domains bounded by confocal conics. Mathematical billiards have applications in any situation that involves collisions and reflections, and any phenomenon that includes reflections and collisions can be modelled using mathematical billiards. This project aims to revolutionise the analysis of billiards within domains bounded by several confocal conics by exploring the relations of such billiards with polygonal billiards, and making research advances with the higher-dimensional generalisations within confocal quadrics and their relations with billiards within polyhedra. The project will link several significant areas of scientific work including polygonal billiards, classical integrable systems, Teichmuller spaces, and relativity theory. The project outcomes will have impact across areas of mathematics such as geometry, algebraic geometry, and dynamical systems.Read moreRead less
Expanding and linking random matrix theory. Fundamental to random matrix theory are certain universality laws, holding in scaling limits to infinite matrix size. A basic question is to quantify the rate of convergence to the universal laws. The analysis of data for the Riemann zeros from prime number theory, and of the spectral form factor probe of chaos in black hole physics, are immediate applications. An analysis involving integrable structures holding for finite matrix size and their asympt ....Expanding and linking random matrix theory. Fundamental to random matrix theory are certain universality laws, holding in scaling limits to infinite matrix size. A basic question is to quantify the rate of convergence to the universal laws. The analysis of data for the Riemann zeros from prime number theory, and of the spectral form factor probe of chaos in black hole physics, are immediate applications. An analysis involving integrable structures holding for finite matrix size and their asymptotics is proposed, allowing the rate to be quantified for a large class of model
ensembles, and providing predictions in the various applied settings. The broad project is to be networked with researchers in the Asia-Oceania region, with the aim of establishing leadership status for Australia.Read moreRead less
Geometric analysis of nonlinear systems. Modern science derives its power from mathematics. The project aims to capture, identify and describe pivotal, transcendental solutions of nonlinear systems that are universal in science, in the sense that they always arise as mathematical models under certain physical limits. The project expects to produce new mathematical methods to describe such functions by using a newly discovered geometric framework. Expected outcomes include the description of elus ....Geometric analysis of nonlinear systems. Modern science derives its power from mathematics. The project aims to capture, identify and describe pivotal, transcendental solutions of nonlinear systems that are universal in science, in the sense that they always arise as mathematical models under certain physical limits. The project expects to produce new mathematical methods to describe such functions by using a newly discovered geometric framework. Expected outcomes include the description of elusive solutions of discrete and higher-dimensional nonlinear systems. This should provide significant benefits, such as new mathematical knowledge, innovative techniques, enhanced scientific capacity in Australia.Read moreRead less