Quantum Spectra. Fundamental quantum processes will play a key role in emerging technologies in the twenty-first century across diverse industries including quantum information technology, quantum computers and electronics, quantum optics, nanoscale quantum microscopes and superconductor technology. Australia has a strong base of expertise in the underpinning quantum disciplines. This project in strategic basic research within mathematical physics will develop a comprehensive and consistent math ....Quantum Spectra. Fundamental quantum processes will play a key role in emerging technologies in the twenty-first century across diverse industries including quantum information technology, quantum computers and electronics, quantum optics, nanoscale quantum microscopes and superconductor technology. Australia has a strong base of expertise in the underpinning quantum disciplines. This project in strategic basic research within mathematical physics will develop a comprehensive and consistent mathematical description of quantum processes. This research will lead to a deeper understanding of quantum processes, keep Australia at the leading edge of international developments and increase Australia's capacity to develop and implement these new technologies.Read moreRead less
Energy, Cosmic Censorship and Black Hole Stability. Human progress is achieved by confronting fundamental questions, at the leading edge of knowledge. This project will lead to better understanding of space-time physics, and of the properties of singular solutions of non-linear hyperbolic equations. Such equations govern a wide range of physical phenomena, including fluid flow, weather and electromagnetic fields.
Discovery Early Career Researcher Award - Grant ID: DE160100250
Funder
Australian Research Council
Funding Amount
$299,436.00
Summary
Advanced methods in combinatorial geometry. This project aims to harness new techniques to solve some challenging open problems related to visibility among sets of points. Combinatorial geometry is the mathematical study of the structure of arrangements of points, lines and other geometric objects in space. Many modern technologies require computation with such geometric data, from computer graphics to robotics and computer vision. Advances in the computational techniques that these technologies ....Advanced methods in combinatorial geometry. This project aims to harness new techniques to solve some challenging open problems related to visibility among sets of points. Combinatorial geometry is the mathematical study of the structure of arrangements of points, lines and other geometric objects in space. Many modern technologies require computation with such geometric data, from computer graphics to robotics and computer vision. Advances in the computational techniques that these technologies use are underpinned by mathematical theory. The last five years has seen major breakthroughs in combinatorial geometry, along with the development of ground-breaking new techniques. Solutions to current problems using these techniques are likely to lead to further theoretical advances and insights.Read moreRead less
Matchings in Combinatorial Structures. The theory of matching in graphs concerns the problem of pairing up objects, subject to constraints on which objects may be paired. It is a well-developed theory that is not only of tremendous mathematical importance, but is also widely applied to efficiently deal with allocation and scheduling problems. Much less is known, however, about the equally important but harder problem of dividing objects into collections of three or more. This project aims to add ....Matchings in Combinatorial Structures. The theory of matching in graphs concerns the problem of pairing up objects, subject to constraints on which objects may be paired. It is a well-developed theory that is not only of tremendous mathematical importance, but is also widely applied to efficiently deal with allocation and scheduling problems. Much less is known, however, about the equally important but harder problem of dividing objects into collections of three or more. This project aims to address this deficiency by developing the theory of matching in important combinatorial objects. The problems it expects to solve are of great significance in their own right, and when considered together may help to lay a foundation for a more general theory of matching.Read moreRead less
Edge decomposition of dense graphs. This project aims to address the edge decomposition of dense graphs, including the Nash-Williams conjecture. Edge decomposition of graphs is important for the mathematical fields of graph theory, combinatorial design theory and finite geometry, and also has strong applications to digital communication and information technologies. It is anticipated that the project will result in methods for edge decomposition of dense graphs, the solution of famous open probl ....Edge decomposition of dense graphs. This project aims to address the edge decomposition of dense graphs, including the Nash-Williams conjecture. Edge decomposition of graphs is important for the mathematical fields of graph theory, combinatorial design theory and finite geometry, and also has strong applications to digital communication and information technologies. It is anticipated that the project will result in methods for edge decomposition of dense graphs, the solution of famous open problems, and a deeper, more cohesive understanding of edge decomposition.Read moreRead less
Hadwiger's graph colouring conjecture. Networks are a pervasive part of modern life. This project seeks to answer one of the deepest unsolved problems in the mathematics of networks, namely Hadwiger's Conjecture. This 65-year old problem suggests a sweeping generalisation of the famous map four-colour theorem, and is at the frontier of research in pure mathematics.
Extremal problems in hypergraph matchings. Matchings in hypergraphs are a way of understanding complex relationships between objects in any set. This project will develop a mathematical theory that covers both extreme and typical cases. This theory will have applications wherever hypergraphs are used as models, for example in machine learning, game theory, databases, data mining and optimisation.
Graph colouring via entropy compression. Graphs and hypergraphs are mathematical structures that model networks. Colouring graphs and hypergraphs is a key problem in many fields including scheduling, computing derivatives, cryptography, and coding theory. This project will apply a revolutionary method called "entropy compression" to produce new mathematical tools and algorithms for colouring graphs and hypergraphs. These results will have significant ramifications for the above applications, and ....Graph colouring via entropy compression. Graphs and hypergraphs are mathematical structures that model networks. Colouring graphs and hypergraphs is a key problem in many fields including scheduling, computing derivatives, cryptography, and coding theory. This project will apply a revolutionary method called "entropy compression" to produce new mathematical tools and algorithms for colouring graphs and hypergraphs. These results will have significant ramifications for the above applications, and will also be of fundamental importance in graph theory itself.Read moreRead less
New approaches to the random generation of combinatorial objects. This project intends to create new methods to generate networks. Networks are useful theoretical combinatorial objects, which also occur in natural contexts such as social networks or the world wide web. Some studies require network simulations and often the best option for this is to generate the networks uniformly at random. Then each one of a given size occurs with the same probability. This project plans to introduce new appro ....New approaches to the random generation of combinatorial objects. This project intends to create new methods to generate networks. Networks are useful theoretical combinatorial objects, which also occur in natural contexts such as social networks or the world wide web. Some studies require network simulations and often the best option for this is to generate the networks uniformly at random. Then each one of a given size occurs with the same probability. This project plans to introduce new approaches to performing this task, enabling it to be done efficiently for a much broader range of networks, graphs and other combinatorial objects than before. This would permit better simulations and efficient and truthful sampling, giving better prediction of network properties and better analysis of algorithms applied to networks.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE170100789
Funder
Australian Research Council
Funding Amount
$324,499.00
Summary
Advances in graph Ramsey theory. This project aims to solve significant questions at the forefront of graph Ramsey theory, which provides the theoretical background for understanding networks that are omnipresent in the modern world. Major progress is anticipated on the recently introduced concept of Ramsey equivalence, including the development of deep new tools that combine probabilistic methods, extremal graph theory and graph decomposition techniques. The project will use these new tools to ....Advances in graph Ramsey theory. This project aims to solve significant questions at the forefront of graph Ramsey theory, which provides the theoretical background for understanding networks that are omnipresent in the modern world. Major progress is anticipated on the recently introduced concept of Ramsey equivalence, including the development of deep new tools that combine probabilistic methods, extremal graph theory and graph decomposition techniques. The project will use these new tools to solve old questions on the structure of minimal Ramsey graphs, thus fostering the international competitiveness of Australian research and enhancing Australia's reputation as a knowledge nation.Read moreRead less