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
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
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
New directions in extremal and structural graph theory. This project aims to attack unsolved problems at the intersection of extremal and structural graph theory, two of the most significant branches of graph theory. Graph theory, which is the mathematics of networks, models many real-world problems and is a key component of modern mathematics. This project expects to develop a theory that synthesises the latest developments in the two fields. It is expected that the tools developed will be wide ....New directions in extremal and structural graph theory. This project aims to attack unsolved problems at the intersection of extremal and structural graph theory, two of the most significant branches of graph theory. Graph theory, which is the mathematics of networks, models many real-world problems and is a key component of modern mathematics. This project expects to develop a theory that synthesises the latest developments in the two fields. It is expected that the tools developed will be widely applicable, for example, in algorithms for network optimisation. The project will build collaborations between Australian researchers and world-leading international mathematicians, and will provide advanced training for talented young researchers.Read moreRead less
Enumeration and properties of large discrete structures. This project aims to study a fundamental property of random graphs, by further developing a recently introduced approach to the problem of enumerating graphs with given degrees. Using this new method, the project expects to generate new knowledge on the number of connections that each node has with other nodes in a random graph, and to develop new strategies for counting the graphs or networks with a given property. The project expects to ....Enumeration and properties of large discrete structures. This project aims to study a fundamental property of random graphs, by further developing a recently introduced approach to the problem of enumerating graphs with given degrees. Using this new method, the project expects to generate new knowledge on the number of connections that each node has with other nodes in a random graph, and to develop new strategies for counting the graphs or networks with a given property. The project expects to produce new theoretical results as well as enhanced capabilities of mathematical research. Potential benefits arise through the uses of these theoretical combinatorial objects to study naturally occurring networks such as social networks, the network of the world wide web, and chemical compounds.Read moreRead less
Enumeration and random generation of contingency tables with given margins. This project aims to find algorithms to construct random tables of numbers having given totals across the rows and down the columns. The aim is also to study properties of such tables. A significant aspect of the project is that it is expected to cover scenarios where all existing methods fail, by deploying recently developed powerful techniques used for random networks in combinatorics. Expected outcomes of this project ....Enumeration and random generation of contingency tables with given margins. This project aims to find algorithms to construct random tables of numbers having given totals across the rows and down the columns. The aim is also to study properties of such tables. A significant aspect of the project is that it is expected to cover scenarios where all existing methods fail, by deploying recently developed powerful techniques used for random networks in combinatorics. Expected outcomes of this project include the development of efficient algorithms that can be used in statistics for identifying relationships between variables in large data sets. This would help bring Australia to the forefront of research in an area that is significant both in data analysis and in discrete mathematics.
Read moreRead less
Triangulations: linking geometry and topology with combinatorics. Triangulations are the method of choice to represent geometric objects given by a finite sample of points. Prominent examples include the pictures produced by the finite element method, polytopes in optimisation, or surfaces in computer graphics.
Knowledge about the triangulations of an object and how they relate to each other is essential for these applications. Seemingly canonical and straightforward methods perform well - or n ....Triangulations: linking geometry and topology with combinatorics. Triangulations are the method of choice to represent geometric objects given by a finite sample of points. Prominent examples include the pictures produced by the finite element method, polytopes in optimisation, or surfaces in computer graphics.
Knowledge about the triangulations of an object and how they relate to each other is essential for these applications. Seemingly canonical and straightforward methods perform well - or not at all, depending on intricate and highly involved mathematical properties.
In this project we combine geometric and topological viewpoints to tackle high-profile questions about triangulations. This will unlock the full potential of combinatorial methods and practical algorithms in applications.Read moreRead less
A unified approach to the design of minimum length networks. This project aims to develop a new approach to designing minimum length interconnection networks by analysing their geometric structure. These networks form the basis of communication, power and transport systems. Optimising the design of such networks is a mathematically challenging problem of high computational complexity. This project will use an innovative method based on a relationship between the geometry of networks and a type o ....A unified approach to the design of minimum length networks. This project aims to develop a new approach to designing minimum length interconnection networks by analysing their geometric structure. These networks form the basis of communication, power and transport systems. Optimising the design of such networks is a mathematically challenging problem of high computational complexity. This project will use an innovative method based on a relationship between the geometry of networks and a type of partitioning of the plane called an oriented Voronoi diagram. The outcome will be efficient new algorithms for designing physical networks, which, in practice, will ultimately lead to a reduction in network infrastructure costs for industries in Australia.
Read moreRead less
Diagram categories and transformation semigroups. A structural understanding of diagram categories is essential in many branches of mathematics and science. Despite this, very few methods for studying such categories are available, a fact this pure mathematics project seeks to rectify. By building strong bridges between diagram categories and semigroup theory, a field of abstract algebra that models transformation and change, the structure of diagram categories may be unlocked with powerful semi ....Diagram categories and transformation semigroups. A structural understanding of diagram categories is essential in many branches of mathematics and science. Despite this, very few methods for studying such categories are available, a fact this pure mathematics project seeks to rectify. By building strong bridges between diagram categories and semigroup theory, a field of abstract algebra that models transformation and change, the structure of diagram categories may be unlocked with powerful semigroup tools developed by the applicant investigator. Diagrammatic insights will also yield new ways to study semigroups, and the many other mathematical structures they interact with. Outcomes will have a lasting impact on both theories as well as the many fields influenced by them.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE200101045
Funder
Australian Research Council
Funding Amount
$330,756.00
Summary
Enhanced methods for approximating the structure of large networks. This project aims to explain fundamental structural features of real-world networks such as the internet and online social networks, by advancing complex-analytical techniques. Current knowledge of properties such as reliability, robustness and optimal allocation of resources rely on assumptions that are invalid in real applications. The project expects to improve understanding of inhomogeneous network models by introducing an i ....Enhanced methods for approximating the structure of large networks. This project aims to explain fundamental structural features of real-world networks such as the internet and online social networks, by advancing complex-analytical techniques. Current knowledge of properties such as reliability, robustness and optimal allocation of resources rely on assumptions that are invalid in real applications. The project expects to improve understanding of inhomogeneous network models by introducing an innovative idea of high-order approximations to complex random settings. Expected outcomes include new tools for approximate counting of discrete objects satisfying given constraints. Applications of these tools could have far-reaching benefits to researchers who study quantitative characteristics of discrete systems.Read moreRead less