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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Homotopy theory: interactions with representation theory and moduli spaces. This proposal will involve young researchers and train them for problem solving in many fields, including management, the sciences, the financial industries, and the development of technologies. Furthermore, many of the projects in this proposal are collaborative and interdisciplinary. It is the CI's sincere hope that this proposal can help bolster communication amongst the wealth of topology, number theory, and mathe ....Homotopy theory: interactions with representation theory and moduli spaces. This proposal will involve young researchers and train them for problem solving in many fields, including management, the sciences, the financial industries, and the development of technologies. Furthermore, many of the projects in this proposal are collaborative and interdisciplinary. It is the CI's sincere hope that this proposal can help bolster communication amongst the wealth of topology, number theory, and mathematical physics experts in Australia. The research in these exciting areas of mathematics will contribute to maintaining Australia's position as a research leader in pure mathematics.
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Broadening Choice and Increasing Diversity in Public Schools. Currently, most families are limited to the public school in their catchment area, meaning the area in which they can afford to live. This leads to socio-economically and ethnically homogenous schools and entrenches disadvantage, as well as denying students the crucial life lessons that flow from being part of a diverse student body. This project aims to investigate a model for allocating public school places that integrates catchment ....Broadening Choice and Increasing Diversity in Public Schools. Currently, most families are limited to the public school in their catchment area, meaning the area in which they can afford to live. This leads to socio-economically and ethnically homogenous schools and entrenches disadvantage, as well as denying students the crucial life lessons that flow from being part of a diverse student body. This project aims to investigate a model for allocating public school places that integrates catchment areas. The expected outcome would be a system that gives families a wider choice, enabling them to enrol in out-of-area schools, while ensuring that allocations remain fair, equitable and balanced, and also delivering benefits such as achieving a desired level of diversity in student populations within schoolsRead moreRead less
Social Network Analysis: Social Media, Peer Effects and the Environment. The aims of this proposal are to better understand the role of networks in different activities such as social media, education, crime and environment-friendly behaviour. The project expects to help inform the design and practice of policies for education and environmental authorities, police and media markets. Social networks are pervasive in Australia. The project tackles issues of criminal gangs in Australian cities, the ....Social Network Analysis: Social Media, Peer Effects and the Environment. The aims of this proposal are to better understand the role of networks in different activities such as social media, education, crime and environment-friendly behaviour. The project expects to help inform the design and practice of policies for education and environmental authorities, police and media markets. Social networks are pervasive in Australia. The project tackles issues of criminal gangs in Australian cities, the political system and environment-friendly behaviours. This project is at the frontier of work in the economics of networks, with expected outcomes to include new models and methods to better understand the impact of social networks. Benefits include clear policy recommendations to improve welfare in Australian society.Read moreRead less
New methods in network economics to study environment-friendly behaviours. This project aims to develop two new methodologies for measuring how people interact with each other and how one’s peers affect their outcomes. The project expects to test these new ground-breaking models for investigating the effect of peers and networks on environmental issues, such as recycling behaviours. The anticipated outcomes of this project include new theoretical and empirical advancements for studying the econo ....New methods in network economics to study environment-friendly behaviours. This project aims to develop two new methodologies for measuring how people interact with each other and how one’s peers affect their outcomes. The project expects to test these new ground-breaking models for investigating the effect of peers and networks on environmental issues, such as recycling behaviours. The anticipated outcomes of this project include new theoretical and empirical advancements for studying the economics of networks and peers for better policy design. Benefits include clear policy recommendations to motivate environment-friendly behaviours. Read moreRead less
Games and decisions with bounded rationality: theory and economic implications. This project will develop concepts and tools for decision making in complex environments, where it is impossible to fully characterise the possible outcomes and factors that may affect them. A central goal will be to integrate heuristic rules such as the precautionary principle with the more formal approach adopted in benefit-cost analysis.
Explicit Construction of Global Function Fields with Many Rational Places. The use of error-correcting codes and cryptosystems is fundamental to the secure and reliable operation of many technological devices that we depend upon in our everyday lives. Essentially invisible, both coding theory and cryptography are essential for banking (ATM machines, e-banking), commerce (e-commerce), defense (cryptography) and entertainment (digital TV and radio, music CDs, DVDs). While certain families of "goo ....Explicit Construction of Global Function Fields with Many Rational Places. The use of error-correcting codes and cryptosystems is fundamental to the secure and reliable operation of many technological devices that we depend upon in our everyday lives. Essentially invisible, both coding theory and cryptography are essential for banking (ATM machines, e-banking), commerce (e-commerce), defense (cryptography) and entertainment (digital TV and radio, music CDs, DVDs). While certain families of "good" codes and cryptosystems can be constructed from specific function fields whose existence is guaranteed by abstract theory, often no actual construction for the function field is currently known. We aim to close this gap, making a greater range of "good" codes and cryptosystems available for practical applications.
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New constructions and techniques for tensor categories. The goal of this project is to make fundamental advances in the structure theory of tensor categories. Such categories play crucial roles in numerous fields of mathematics, physics and beyond. New methods, theory and examples will be developed, inspired by algebra, representation theory and geometry. These will then be applied in the foundational study of tensor categories for (dis)proving several of the most important open conjectures in t ....New constructions and techniques for tensor categories. The goal of this project is to make fundamental advances in the structure theory of tensor categories. Such categories play crucial roles in numerous fields of mathematics, physics and beyond. New methods, theory and examples will be developed, inspired by algebra, representation theory and geometry. These will then be applied in the foundational study of tensor categories for (dis)proving several of the most important open conjectures in the field. This will open new perspectives for applications in other areas, most notably in representation theory. Other benefits include enhanced international collaboration and scientific capacity in Australia.Read moreRead less
New dualities in modular representation theory. This project aims to introduce new geometric techniques in the modular representation theory of the general linear group, and resolve fundamental questions about modular representations of orthogonal and symplectic groups. Representation theory, the mathematical study of symmetry, has applications in many areas ranging from physics to computer science. The modular setting, where the characteristic of the field is positive, is a source of the deepes ....New dualities in modular representation theory. This project aims to introduce new geometric techniques in the modular representation theory of the general linear group, and resolve fundamental questions about modular representations of orthogonal and symplectic groups. Representation theory, the mathematical study of symmetry, has applications in many areas ranging from physics to computer science. The modular setting, where the characteristic of the field is positive, is a source of the deepest conjectures in this subject. Recent breakthroughs have provided approaches to some problems in modular representation theory that were previously inaccessible. This project will promote research at the forefront of the field, and help maintain Australia's international standing. It will train Australian postgraduates, who will be able to apply their research skills in industry, and strengthen the country's ties to a vibrant international community.Read moreRead less
Asymptotic Expansions and Large Deviations in Probability and Statistics: Theory and Applications. Statistics is the major enabling science in a number of disciplines. This is fundamental research in probability and statistics but it has wide applications in Biology and Social Sciences which will ultimately be of national benefit. The behaviour of self normalized sums is an exciting new area of fundamental research that has implications for the application of statistics in many areas. U-statist ....Asymptotic Expansions and Large Deviations in Probability and Statistics: Theory and Applications. Statistics is the major enabling science in a number of disciplines. This is fundamental research in probability and statistics but it has wide applications in Biology and Social Sciences which will ultimately be of national benefit. The behaviour of self normalized sums is an exciting new area of fundamental research that has implications for the application of statistics in many areas. U-statistics for dependent situations has direct application to understanding financial time series and the analysis of sample survey data. Saddlepoint methods provide extremely accurate approximations in a number of important applications.
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