Sparse grid approximations and fitting using generalised combination techniques. Sparse grid techniques provide an effective tool to deal with the
computational curse of dimensionality which is a constant challenge in
modelling complex data. The proposed research is aimed at the
development and analysis of algorithms for data fitting with sparse
grids using variants of the combination technique. The outcome of the
research is a theory which will provide insights in the applicability,
limit ....Sparse grid approximations and fitting using generalised combination techniques. Sparse grid techniques provide an effective tool to deal with the
computational curse of dimensionality which is a constant challenge in
modelling complex data. The proposed research is aimed at the
development and analysis of algorithms for data fitting with sparse
grids using variants of the combination technique. The outcome of the
research is a theory which will provide insights in the applicability,
limitations and the convergence properties of the proposed
algorithms. The outcomes will be widely applicable in modelling of
large scale and complex data as is encountered in areas of
bioinformatics, physics and experimental studies of complex systems.
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Solving inverse problems with Iterative regularisation and convex penalties. This project aims to develop and investigate new computational procedures for the solution of inverse problems which do not have the usual smoothness properties (or source conditions) required for the traditional regularisation methods. Examples of such inverse problems are very common and include image restoration, photo-acoustic tomography and spectroscopy. It is anticipated that this project will substantially extend ....Solving inverse problems with Iterative regularisation and convex penalties. This project aims to develop and investigate new computational procedures for the solution of inverse problems which do not have the usual smoothness properties (or source conditions) required for the traditional regularisation methods. Examples of such inverse problems are very common and include image restoration, photo-acoustic tomography and spectroscopy. It is anticipated that this project will substantially extend the toolbox of methods for such problems utilising ideas from Banach spaces, convex analysis, parallel computing and optimisation. This project is expected to make a substantial contribution to a better understanding of inverse problems and their solution procedures.Read moreRead less
Theory and applications of three dimensional fractal transformations. The purpose of this project is to develop the theory and algorithms for a new class of continuous mappings between fractals. Outcomes include a better understanding of fractals, substantially better algorithms for fractal compression and many new applications.
Algebraic K-theory and groups. This project will study the K-theory of division algebras, their generalisation to Azumaya algebras and the nonstable K-theory of rings. Expected outcomes would enhance our understanding on the structure of these K groups.
The goal is to settle some of the most significant conjectures in the subject: Bak's solvability of nonstable K groups over rings and the Merkurjev-Suslin conjectures on reduced K theory of division rings.
The study of these problems contribu ....Algebraic K-theory and groups. This project will study the K-theory of division algebras, their generalisation to Azumaya algebras and the nonstable K-theory of rings. Expected outcomes would enhance our understanding on the structure of these K groups.
The goal is to settle some of the most significant conjectures in the subject: Bak's solvability of nonstable K groups over rings and the Merkurjev-Suslin conjectures on reduced K theory of division rings.
The study of these problems contributes to and draws from such topics as group theory, commutative ring theory, algebraic number theory and algebraic geometry.
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Categorical symmetries in representation theory. This project aims to develop categorical symmetries of central objects in mathematics such as braid groups, the Hilbert scheme of points, and the Virasoro algebra. The concept of symmetry is an important organising principle in science. Representation theory is the field of mathematics concerned with studying symmetries. The problems proposed have connections to many different areas including algebra, geometry, topology, and mathematical physics. ....Categorical symmetries in representation theory. This project aims to develop categorical symmetries of central objects in mathematics such as braid groups, the Hilbert scheme of points, and the Virasoro algebra. The concept of symmetry is an important organising principle in science. Representation theory is the field of mathematics concerned with studying symmetries. The problems proposed have connections to many different areas including algebra, geometry, topology, and mathematical physics. This project expects to advance pure mathematics and provide potential benefit in many related fields.Read moreRead less
Low dimensional categories. This project plans to study fundamental examples of higher categories in dimensions 2, 3, and 4, with the goal of understanding their essential features and building appropriate tools and theoretical frameworks for working with them. This work would have applications in several areas of mathematics including representation theory, low dimensional topology and topological quantum computing. Higher categories let us study the possible shapes of space (dimensions 2, 3, a ....Low dimensional categories. This project plans to study fundamental examples of higher categories in dimensions 2, 3, and 4, with the goal of understanding their essential features and building appropriate tools and theoretical frameworks for working with them. This work would have applications in several areas of mathematics including representation theory, low dimensional topology and topological quantum computing. Higher categories let us study the possible shapes of space (dimensions 2, 3, and 4, are the relevant ones for the world we live in), and also the dimensions in which we find the most interesting examples. The project plans to investigate particular examples related to exceptional Lie algebras, fusion categories, and categorical link invariants.Read moreRead less
Topics in triangulated categories. This project in pure mathematics, more specifically in modern homological algebra, builds on work started by the chief investigator in the last five years. What has already been done has achieved striking results, solving very different problems that have been open for two decades. And there seem to be many directions in which it could be pursued further.
The international mathematical community seems intrigued by what the chief investigator has achieved recen ....Topics in triangulated categories. This project in pure mathematics, more specifically in modern homological algebra, builds on work started by the chief investigator in the last five years. What has already been done has achieved striking results, solving very different problems that have been open for two decades. And there seem to be many directions in which it could be pursued further.
The international mathematical community seems intrigued by what the chief investigator has achieved recently - judging by invitations to give prestigious talks and the feedback at these events. The expected outcome is major progress in our understanding of derived categories, as well as diverse applications. The benefit will be to enhance the international stature of Australian science.Read moreRead less
Moduli, invariants, and algebraisation. This project is in pure mathematics. It aims to address gaps in our
knowledge in the modern geometries and their associated algebraic structures that arise in classification problems that pervade mathematics and its applications.
This project expects to generate new knowledge in modern algebra and geometry.
Expected outcomes of this project include major progress in our
understanding of invariants of derived categories of algebraic stacks and the
relat ....Moduli, invariants, and algebraisation. This project is in pure mathematics. It aims to address gaps in our
knowledge in the modern geometries and their associated algebraic structures that arise in classification problems that pervade mathematics and its applications.
This project expects to generate new knowledge in modern algebra and geometry.
Expected outcomes of this project include major progress in our
understanding of invariants of derived categories of algebraic stacks and the
relationship between algebraic and other geometries.
The benefit will be to enhance the international stature of Australian science.Read moreRead less
Banking System Competition and the Macro-economy. Australia has one of the most concentrated banking sectors in the world, generating concerns regarding its efficiency. This project aims to develop unified frameworks to understand and evaluate quantitatively how the structure of the banking industry affects the macro-economy and provide policy recommendations for establishing a healthy and efficient banking industry. This project expects to improve understanding of the welfare trade-off between ....Banking System Competition and the Macro-economy. Australia has one of the most concentrated banking sectors in the world, generating concerns regarding its efficiency. This project aims to develop unified frameworks to understand and evaluate quantitatively how the structure of the banking industry affects the macro-economy and provide policy recommendations for establishing a healthy and efficient banking industry. This project expects to improve understanding of the welfare trade-off between bank competition and economic well-being to enable policymakers to better determine the optimal concentration of banking sector in Australia. This will enhance the productivity and international competitiveness of Australia’s financial system and the broader economy.Read moreRead less
Ethics and risk. This project aims to develop a theory of risk. From the extreme to the everyday, from warfare to the drive to work, the modern world is unimaginable without mutual imposition of risk. Philosophers must explain how risks can be justified, or risk irrelevance. This project will use the tools of ethics (the study of right and wrong action) and decision theory (the study of rational decision-making under uncertainty) to develop a comprehensive theory of the ethics of risk. This proj ....Ethics and risk. This project aims to develop a theory of risk. From the extreme to the everyday, from warfare to the drive to work, the modern world is unimaginable without mutual imposition of risk. Philosophers must explain how risks can be justified, or risk irrelevance. This project will use the tools of ethics (the study of right and wrong action) and decision theory (the study of rational decision-making under uncertainty) to develop a comprehensive theory of the ethics of risk. This project is expected to improve understanding of the risks people impose on others as individuals and as a society.Read moreRead less