Discrete integrable systems. Discrete integrable systems are a fundamental generalisation of traditional integrable systems. This project, combining 5 world experts from 3 countries and 2 early career researchers, will expand and systematise this new interdisciplinary field, and will place Australia at the forefront of this intensive international activity.
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
Multi-dimensionally consistent integrable systems in geometry and algebra. This project aims to address in an innovative manner a long-standing open problem in nonlinear mathematics, namely the determination of the algebraic and geometric origin of integrable systems. It is expected to make a fundamental contribution towards integrable systems theory. The latter provides unique access to the analytic treatment of nonlinear phenomena not only in physics but also a remarkably diverse range of area ....Multi-dimensionally consistent integrable systems in geometry and algebra. This project aims to address in an innovative manner a long-standing open problem in nonlinear mathematics, namely the determination of the algebraic and geometric origin of integrable systems. It is expected to make a fundamental contribution towards integrable systems theory. The latter provides unique access to the analytic treatment of nonlinear phenomena not only in physics but also a remarkably diverse range of areas in mathematics. Expected outcomes include extended, unified and novel key mathematical concepts in a discrete setting and their applications in algebraic and geometric contexts. Due to the choice of participants, it is anticipated that Australia will benefit from strengthened research collaborations with Germany.Read moreRead less
Australian Laureate Fellowships - Grant ID: FL120100094
Funder
Australian Research Council
Funding Amount
$3,184,657.00
Summary
Geometric construction of critical solutions of nonlinear systems. Whether we are looking at waves on a beach, the dispersal of herds of animals in a landscape, or the interaction of black holes, their patterns of movement rely on rules expressed by non-linear mathematical models. This project will aim to create new mathematical methods to describe the solutions of non-linear systems, which are ubiquitous in modern science.
Critical solutions of nonlinear systems. Whether we are looking at waves on a beach, the dispersal of herds of animals in a landscape, or the interaction of black holes, their patterns of movement rely on rules expressed by nonlinear mathematical models. This project aims to create new mathematical methods to describe critical solutions of nonlinear systems, which are ubiquitous in modern science.
Reflection Groups and Discrete Dynamical Systems. This project aims to solve long-standing problems in discrete dynamical systems that are of particular interest to physics, by using reflection groups to reveal unexpected geometric insights. Mathematics has the power to abstract crucial patterns from complex observations. Symmetries familiar in the real world, like the hexagonal patterns of honeycombs, arise inside convoluted structures in high-dimensional systems. By revealing the structure of ....Reflection Groups and Discrete Dynamical Systems. This project aims to solve long-standing problems in discrete dynamical systems that are of particular interest to physics, by using reflection groups to reveal unexpected geometric insights. Mathematics has the power to abstract crucial patterns from complex observations. Symmetries familiar in the real world, like the hexagonal patterns of honeycombs, arise inside convoluted structures in high-dimensional systems. By revealing the structure of space-filling polytopes in integrable systems, the project seeks to find sought-after reductions of high-dimensional discrete models to two dimensions. Expected outputs include new reductions to discrete Painlevé equations, new circle patterns useful for computer graphics and discrete holomorphic functions.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
Algebraic interpretations of discrete integrable equations. The important mathematical disciplines of discrete geometry on one hand, and structure in discrete non-linear dynamics known as integrability on the other, have an emerging and fruitful interrelation. This project will construct a new algebraic framework in order to better understand and exploit this point of intersection.
Many-body problems. The discovery of new superheavy elements, chemical evolution of the Universe, nuclear reactions deep under the Coulomb barrier in nuclear reactors, in stars and during the Big Bang Nucleosynthesis, accuracy of precise atomic clocks, consistency of the Standard Model in strong fields are among the most vital problems of modern science. This project suggests several new ideas in these areas, which are based on knowledge accumulated in different research fields. The outcomes of ....Many-body problems. The discovery of new superheavy elements, chemical evolution of the Universe, nuclear reactions deep under the Coulomb barrier in nuclear reactors, in stars and during the Big Bang Nucleosynthesis, accuracy of precise atomic clocks, consistency of the Standard Model in strong fields are among the most vital problems of modern science. This project suggests several new ideas in these areas, which are based on knowledge accumulated in different research fields. The outcomes of the research will help Australia to build up a "critical mass" of scientific expertise, which is necessary to place and keep it among leaders in these frontier areas of physics, and to train the next generation of experts in these fields.Read moreRead less
Modelling quantum dynamics of electronic excited states in complex molecular materials. Understanding new materials that are the basis of new sources of renewable energy sources represents a major scientific challenge. Many of these materials are composed of large organic molecules containing hundreds of atoms. Their properties and the concepts needed to understand these materials are distinctly different from semiconductors such as silicon. This research will enhance our ability to design bett ....Modelling quantum dynamics of electronic excited states in complex molecular materials. Understanding new materials that are the basis of new sources of renewable energy sources represents a major scientific challenge. Many of these materials are composed of large organic molecules containing hundreds of atoms. Their properties and the concepts needed to understand these materials are distinctly different from semiconductors such as silicon. This research will enhance our ability to design better materials and optimize the performance of organic solar cells and LEDs. Australia's capacity for research and development in this scientifically challenging and technologically important field will be enhanced by this project. Read moreRead less