Continued Fractions and Torsion on Hyperelliptic Curves. Scientific advance should not blindly add to our knowledge; a true advance brings insights that collapse different issues into one. Understanding more is to need to remember less. For an important class of examples, this project identifies the study of a fundamental invariant of a quadratic number field, its regulator and hence its class number, with maximum torsion on the Jacobian variety of an hyperelliptic curve. The investigator's meth ....Continued Fractions and Torsion on Hyperelliptic Curves. Scientific advance should not blindly add to our knowledge; a true advance brings insights that collapse different issues into one. Understanding more is to need to remember less. For an important class of examples, this project identifies the study of a fundamental invariant of a quadratic number field, its regulator and hence its class number, with maximum torsion on the Jacobian variety of an hyperelliptic curve. The investigator's methods will surprise some longstanding problems into submission and in particular will lead them to reveal full data on torsion on hyperelliptic curves of low genus.
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Asymptotic Geometric Analysis and Machine Learning. Phenomena in large dimensions appear in a number of domains of Mathematics and adjacent domains of science (e.g. Computer Science), dealing with functions of infinitely growing number of parameters. Here, we focus on several questions naturally linked to Asymptotic Geometric Analysis which have natural applications to Statistical Learning Theory. We intend to use geometric, probabilistic and combinatorial methods to investigate these problems, ....Asymptotic Geometric Analysis and Machine Learning. Phenomena in large dimensions appear in a number of domains of Mathematics and adjacent domains of science (e.g. Computer Science), dealing with functions of infinitely growing number of parameters. Here, we focus on several questions naturally linked to Asymptotic Geometric Analysis which have natural applications to Statistical Learning Theory. We intend to use geometric, probabilistic and combinatorial methods to investigate these problems, with an emphasis on modern tools in Empirical Processes Theory and the theory of Random Matrices.Read moreRead less
Algorithmics for Interactive 2.5D Graph Drawing. Technological advances have provided a data deluge over the past few years, and consequently have led to many large and complex network models in many domains. This includes terrorrist networks and biological networks, software engineering structures, and webgraphs. Visualisation is an effective tool in helping humans to understand such networks. This project aims to provide a new direction in network visualisation, using 2.5 dimensions. The algor ....Algorithmics for Interactive 2.5D Graph Drawing. Technological advances have provided a data deluge over the past few years, and consequently have led to many large and complex network models in many domains. This includes terrorrist networks and biological networks, software engineering structures, and webgraphs. Visualisation is an effective tool in helping humans to understand such networks. This project aims to provide a new direction in network visualisation, using 2.5 dimensions. The algorithms developed in the project will help security analysts to detect abnormal behavious such as money laundering, help biologists understand protein-protein interaction networks, and help engineers to understand large software systems.Read moreRead less
Efficient Pre-Processing of Hard Problems: New Approaches, Basic Theory and Applications. Computers store even larger amounts of data about all aspects of human and industrial activity. However, they have not become significantly better at solving common problems in optimization and search. Traditional complexity theory indicates many of these problems require algorithms that are very unlikely to exist. The Parameterized Complexity approach allows us to obtain very efficient algorithms for a lar ....Efficient Pre-Processing of Hard Problems: New Approaches, Basic Theory and Applications. Computers store even larger amounts of data about all aspects of human and industrial activity. However, they have not become significantly better at solving common problems in optimization and search. Traditional complexity theory indicates many of these problems require algorithms that are very unlikely to exist. The Parameterized Complexity approach allows us to obtain very efficient algorithms for a large variety of problems, but the machinery required was diverse and complicated. This research will organize the machinery into a new approach that systematically finds good algorithms by applying simplifications around a parameter of the domain of the problem. As a result, efficient algorithms are obtained for many diverse areas.Read moreRead less
Optimal Transforms of Random Vectors. This proposal focusses on development of optimal transforms to describe and model nonlinear phenomena when only statistical information is known. An optimal transform is a mathematical procedure that enables us to process information in a way that is most suited to the task in hand. These transforms have been successfully used in approximation, information theory, communications, control theory and signal and image processing. Applications include modelli ....Optimal Transforms of Random Vectors. This proposal focusses on development of optimal transforms to describe and model nonlinear phenomena when only statistical information is known. An optimal transform is a mathematical procedure that enables us to process information in a way that is most suited to the task in hand. These transforms have been successfully used in approximation, information theory, communications, control theory and signal and image processing. Applications include modelling of physical, chemical and biological systems, filtering and compression of signals and data classification and clustering. We propose two new hybrid models for realistic transforms in a general structural framework.
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Fast, practical and effective algorithms for clustering with advice. To maintain a safe and healthy society, government and industry need high quality immunization and national security databases. Since we cannot afford to have duplicate, incomplete and conflicting records that refer to the same person, we unify them by identifying clusters of related records.
In the emerging field of functional genomics, diagnosis of certain diseases is enhanced by determining which genes act together. Diffe ....Fast, practical and effective algorithms for clustering with advice. To maintain a safe and healthy society, government and industry need high quality immunization and national security databases. Since we cannot afford to have duplicate, incomplete and conflicting records that refer to the same person, we unify them by identifying clusters of related records.
In the emerging field of functional genomics, diagnosis of certain diseases is enhanced by determining which genes act together. Different experimental runs might result in different clusterings of genes: we need one consensus clustering that summarizes the experimental outcomes.
Cleaning databases and combining clusterings by hand would require vast amounts of time. This project will result in faster and more accurate computational procedures.Read moreRead less
Modelling, Identification and Control of Complex Networks. Australia has been well known for its leading research in systems and control and many real-world applications in, for instance, telecommunications, defence, power grids and life sciences. This project will further promote Australia's leading position in the emerging new research field - complex networks by theoretical breakthrough in modelling, identification and control of complex networks, and cutting-edge platform technology that can ....Modelling, Identification and Control of Complex Networks. Australia has been well known for its leading research in systems and control and many real-world applications in, for instance, telecommunications, defence, power grids and life sciences. This project will further promote Australia's leading position in the emerging new research field - complex networks by theoretical breakthrough in modelling, identification and control of complex networks, and cutting-edge platform technology that can help Australian energy industry to reduce greenhouse emissions. It will also result in education of the next generation research leaders in this emerging field.Read moreRead less
Efficient Synchronisation of Large Repositories. Accuracy and maintenance of vast quantities of data are essential for any modern society. The economy, health institutes and industries, and our defence and legal systems rely on having data being distributed widely and securely, and on queries being answered accurately and quickly. Complete synchronisation of databases is often impossible due to the limitations of internet bandwidth. Better compression techniques have the potential to allow crit ....Efficient Synchronisation of Large Repositories. Accuracy and maintenance of vast quantities of data are essential for any modern society. The economy, health institutes and industries, and our defence and legal systems rely on having data being distributed widely and securely, and on queries being answered accurately and quickly. Complete synchronisation of databases is often impossible due to the limitations of internet bandwidth. Better compression techniques have the potential to allow critical data to be distributed much more efficiently; we anticipate in some applications that the size of a compressed file could be reduced tenfold or more compared to previous best methods, leading to dramatic savings.Read moreRead less
A new perturbation method for solving singular operator equations with applications to complex systems. This project will develop new methods for analysis of web-based search routines such as Google PageRank, a new algorithm for optimal estimation of random signals, more accurate error analysis in the approximate solution of singular systems of equations and enhanced understanding of models for the simulated management of urban stormwater. The project will involve collaboration between two Aus ....A new perturbation method for solving singular operator equations with applications to complex systems. This project will develop new methods for analysis of web-based search routines such as Google PageRank, a new algorithm for optimal estimation of random signals, more accurate error analysis in the approximate solution of singular systems of equations and enhanced understanding of models for the simulated management of urban stormwater. The project will involve collaboration between two Australian universities and a leading European Research Institute. It will provide employment and vital training for two postdoctoral Research fellows and research projects for three postgraduate students and two honours students.Read moreRead less
Supporting personalised learning in secondary schools through the use of specific mathematics assessments that reveal thinking. This project will improve numeracy standards in secondary schools, by assisting teachers to personalise the teaching of each student. Through detailed research, the project will develop sophisticated assessments that clearly reveal students' mathematical thinking. On-line diagnosis will suggest teaching activities that can bring about desired conceptual changes. The eff ....Supporting personalised learning in secondary schools through the use of specific mathematics assessments that reveal thinking. This project will improve numeracy standards in secondary schools, by assisting teachers to personalise the teaching of each student. Through detailed research, the project will develop sophisticated assessments that clearly reveal students' mathematical thinking. On-line diagnosis will suggest teaching activities that can bring about desired conceptual changes. The effectiveness of the resources will be thoroughly tested, for impact on student achievement and the mathematics teaching skills of teachers, especially those working out-of-field. Rural schools will have equal access to these resources. Good numeracy provides the foundation for life-long learning, employment satisfaction and the economic competitiveness of Australia.Read moreRead less