Discovery Early Career Researcher Award - Grant ID: DE180100957
Funder
Australian Research Council
Funding Amount
$339,328.00
Summary
Partial differential equations, free boundaries and applications. This project aims to investigate fundamental problems in the analysis of partial differential equations and free boundary theory, to develop advanced mathematical theories with the possibility of important applications. The expected outcome is the establishment of a regularity and classification theory for nonlocal equations and for free boundary problems in linear and nonlinear settings. The benefit of the project lies in a concr ....Partial differential equations, free boundaries and applications. This project aims to investigate fundamental problems in the analysis of partial differential equations and free boundary theory, to develop advanced mathematical theories with the possibility of important applications. The expected outcome is the establishment of a regularity and classification theory for nonlocal equations and for free boundary problems in linear and nonlinear settings. The benefit of the project lies in a concrete advancement of the mathematical research with advantages for a deeper understanding of complex phenomena in physics and biology. Some of the problems also provide results useful for industrial applications.Read moreRead less
Special Research Initiatives - Grant ID: SR0354716
Funder
Australian Research Council
Funding Amount
$10,000.00
Summary
Energetically Open Systems Research Network Study. Conceptual frameworks arising in the physical sciences, such as non-equilibrium statistical mechanics and thermodynamics, synergetics, chaos and dynamical systems theory, are seminal in the emerging science of complexity. This study will lay the groundwork for a network to link Australian and overseas research on these fundamental concepts, and their application within the context of entropy-producing systems vital to the long-term sustainabilit ....Energetically Open Systems Research Network Study. Conceptual frameworks arising in the physical sciences, such as non-equilibrium statistical mechanics and thermodynamics, synergetics, chaos and dynamical systems theory, are seminal in the emerging science of complexity. This study will lay the groundwork for a network to link Australian and overseas research on these fundamental concepts, and their application within the context of entropy-producing systems vital to the long-term sustainability of the earth - oceans, atmosphere, biosphere, CO2-free energy production, space and solar environment. The network would facilitate the development of young investigators and be linked into wider complex systems networks such as the CSIRO Centre for Complex Systems Science.Read moreRead less
Hypergraph models for complex discrete systems. This project aims to better understand the structure and properties of very large hypergraphs of various kinds. Hypergraphs are very general mathematical objects which can be used to model complex discrete systems. They arise naturally in many areas such as ecology, chemistry and computer science. Despite this, our theoretical understanding of very large, or random, hypergraphs lags far behind the intensely-studied special case of graphs. This proj ....Hypergraph models for complex discrete systems. This project aims to better understand the structure and properties of very large hypergraphs of various kinds. Hypergraphs are very general mathematical objects which can be used to model complex discrete systems. They arise naturally in many areas such as ecology, chemistry and computer science. Despite this, our theoretical understanding of very large, or random, hypergraphs lags far behind the intensely-studied special case of graphs. This project will answer many fundamental questions about large, random hypergraphs. The expected outcomes of the project also include new tools for working with hypergraphs, such as efficient algorithms for sampling hypergraphs. These outcomes will benefit researchers who use hypergraphs in their work and will enhance Australia's reputation for research in this area.Read moreRead less
Analysis of the structure of latin squares. The benefits to Australia of fundamental research in core disciplines such as mathematics are well documented. Discrete mathematics and combinatorics are boom disciplines of the computer age and this project seeks new knowledge concerning basic building blocks of combinatorial mathematics. The outcomes will be of interest to theoretical discrete mathematicians around the world, enhancing Australia's already high research profile in this important area ....Analysis of the structure of latin squares. The benefits to Australia of fundamental research in core disciplines such as mathematics are well documented. Discrete mathematics and combinatorics are boom disciplines of the computer age and this project seeks new knowledge concerning basic building blocks of combinatorial mathematics. The outcomes will be of interest to theoretical discrete mathematicians around the world, enhancing Australia's already high research profile in this important area of pure mathematical research. Importantly, the problems under investigation offer substantial opportunity for excellent postgraduate training, critical for the future of Australian research. Read moreRead less
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
The Zarankiewicz problem through linear hypergraphs and designs. The Zarankiewicz problem is a famous open problem with deep connections to many different areas of mathematics. Despite continued attention from some of the world's most celebrated mathematicians, it has remained unsolved for over 70 years. This project aims to make major progress on the Zarankiewicz problem by utilising a novel approach based in the field of combinatorial design theory. This approach will leverage recent major bre ....The Zarankiewicz problem through linear hypergraphs and designs. The Zarankiewicz problem is a famous open problem with deep connections to many different areas of mathematics. Despite continued attention from some of the world's most celebrated mathematicians, it has remained unsolved for over 70 years. This project aims to make major progress on the Zarankiewicz problem by utilising a novel approach based in the field of combinatorial design theory. This approach will leverage recent major breakthroughs in design theory concerning edge decompositions of dense hypergraphs.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.