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Discovery Early Career Researcher Award - Grant ID: DE190100666
Funder
Australian Research Council
Funding Amount
$381,000.00
Summary
Extremal combinatorics meets finite geometry. This project aims to investigate important open problems lying at the intersection of two areas of mathematics, extremal combinatorics and finite geometry. The project will focus on the area of discrete mathematics, which has been at the centre of some of recent developments in mathematics and computer science. This project proposes new methods, derived from algebra, geometry and computer science, to tackle important extremal problems in finite geome ....Extremal combinatorics meets finite geometry. This project aims to investigate important open problems lying at the intersection of two areas of mathematics, extremal combinatorics and finite geometry. The project will focus on the area of discrete mathematics, which has been at the centre of some of recent developments in mathematics and computer science. This project proposes new methods, derived from algebra, geometry and computer science, to tackle important extremal problems in finite geometry. The project will provide answers to a number of open problems in extremal combinatorics and finite geometry. Moreover, new methods will be developed which will have an interdisciplinary impact.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
Decompositions of graphs into cycles: Alspach's Conjecture and the Oberwolfach problem. Graph theory is used extensively to model and solve practical problems in physical, biological and social systems. By answering two long-standing and fundamental questions, the project will extend a long tradition of Australian research excellence in the field, and provide substantial high-quality postgraduate training in line with national needs.
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
Fractional decomposition of graphs and the Nash-Williams conjecture. Nash-Williams' conjecture is a famous unsolved problem about decomposing graphs (abstract networks). Breakthrough results achieved in recent years have shown that the conjecture, along with other major graph decomposition problems, could be solved if only more were known about fractional decomposition. This project aims to clear this bottleneck to progress by dramatically expanding the state of knowledge on fractional decomposi ....Fractional decomposition of graphs and the Nash-Williams conjecture. Nash-Williams' conjecture is a famous unsolved problem about decomposing graphs (abstract networks). Breakthrough results achieved in recent years have shown that the conjecture, along with other major graph decomposition problems, could be solved if only more were known about fractional decomposition. This project aims to clear this bottleneck to progress by dramatically expanding the state of knowledge on fractional decomposition. Expected outcomes include major progress on Nash-Williams' conjecture and related graph decomposition problems. This should enhance Australia's research reputation in pure mathematics and provide benefits in downstream applications areas including statistics, data transmission, and fibre-optic networks.Read moreRead less
The Oberwolfach Problem and related Graph Factorisations. Graph factorisation is an active area of research in combinatorial mathematics that is driven both by theoretical questions and by new and varied applications, particularly in digital communication and information technologies. The aim of this project is to solve the Oberwolfach Problem: a fundamental and historically significant graph factorisation question that has intrigued researchers for decades. Building on recent breakthroughs, new ....The Oberwolfach Problem and related Graph Factorisations. Graph factorisation is an active area of research in combinatorial mathematics that is driven both by theoretical questions and by new and varied applications, particularly in digital communication and information technologies. The aim of this project is to solve the Oberwolfach Problem: a fundamental and historically significant graph factorisation question that has intrigued researchers for decades. Building on recent breakthroughs, new and widely applicable graph factorisation techniques are intended to be developed. The project outcomes are expected to have ongoing influence and impact on research in the field.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.