Representations and Applications of Quantum Groups. The theory of quantum groups originated from soluble lattice models in statistical mechanics, but it turned out to have important applications to a wide range of subjects in mathematics and physics. For this reason, quantum groups have occupied a central stage of international mathematical research for the last decade, and continue to attract great interest. This project addresses some important open problems on representations and applicatio ....Representations and Applications of Quantum Groups. The theory of quantum groups originated from soluble lattice models in statistical mechanics, but it turned out to have important applications to a wide range of subjects in mathematics and physics. For this reason, quantum groups have occupied a central stage of international mathematical research for the last decade, and continue to attract great interest. This project addresses some important open problems on representations and applications of some finite dimensional quantum groups.Read moreRead less
Noncommutative Algebraic Geometry. As algebra moves into the twenty-first century, we see a strong trend towards interactions with geometry. This project is right in the thick of this trend and will keep Australia abreast of some of the most interesting developments in algebra. The project seeks to start up a research group in noncommutative algebraic geometry which will foster a lively intellectual atmosphere. This will involve training postgraduate students, inviting international experts to g ....Noncommutative Algebraic Geometry. As algebra moves into the twenty-first century, we see a strong trend towards interactions with geometry. This project is right in the thick of this trend and will keep Australia abreast of some of the most interesting developments in algebra. The project seeks to start up a research group in noncommutative algebraic geometry which will foster a lively intellectual atmosphere. This will involve training postgraduate students, inviting international experts to give seminar talks and establishing relations with other Australian mathematicians in related areas.Read moreRead less
Towards Mike Artin's conjecture. Non-commutative algebra and algebraic geometry are both classical branches of mathematics with much depth to them. As a result, the recent study of the interactions between the two disciplines has proven to be fertile ground for many important developments in mathematics. This project ensures that Australia remains a part of these developments.
Polynomial representations of the Hecke algebra. This project will offer a great opportunity for talented students to engage in internationally competitive research in mathematics. In addition, through international collaboration, this project will be able to deliver an online database with software libraries which will be a world benchmark for computation with multivariate polynomials.
Noncommutative geometry in representation theory and quantum physics. One of the most important problems in natural science is to understand the structure of spacetime at the Planck scale. Mathematical investigations in recent years have predicted that at this scale, spacetime becomes noncommutative. Taking this noncommutativity into account, the project brings together geometry, algebra and quantum mechanics to develop new mathematical theories required for addressing the problem. It promises ....Noncommutative geometry in representation theory and quantum physics. One of the most important problems in natural science is to understand the structure of spacetime at the Planck scale. Mathematical investigations in recent years have predicted that at this scale, spacetime becomes noncommutative. Taking this noncommutativity into account, the project brings together geometry, algebra and quantum mechanics to develop new mathematical theories required for addressing the problem. It promises to make fundamental contributions to both mathematics and theoretical physics. Read moreRead less
The mathematical analysis of ultracold quantum gases. Ongoing developments in the experimental realisation of ultracold quantum
gases play a leading role in the international effort towards the
eventual realisation of quantum technology. This project brings together Australian
researchers with complementary strengths to develop a sophisticated range of innovative
mathematical tools for understanding these fundamental quantum systems. The expected
outcomes will thus include potentially far r ....The mathematical analysis of ultracold quantum gases. Ongoing developments in the experimental realisation of ultracold quantum
gases play a leading role in the international effort towards the
eventual realisation of quantum technology. This project brings together Australian
researchers with complementary strengths to develop a sophisticated range of innovative
mathematical tools for understanding these fundamental quantum systems. The expected
outcomes will thus include potentially far reaching impacts on downstream quantum technology.
The project will contribute to training mathematically talented students and thus take essential
steps to establish the long term future of mathematical physics in Australia. It will also establish enduring key international research links.Read moreRead less
Algebras with Frobenius morphisms and quantum groups. In this digitalized world, our life relies on mathematics more than ever. Counting and numbers are just one example of this. Another is the public key codes for online payments and transactions. Mathematics is of enormous importance in this technology dominated age. This proposal is to carry out high level mathematical research in Australia. Basic research on quantum groups underpins applied research and certain areas such as quantum mechanic ....Algebras with Frobenius morphisms and quantum groups. In this digitalized world, our life relies on mathematics more than ever. Counting and numbers are just one example of this. Another is the public key codes for online payments and transactions. Mathematics is of enormous importance in this technology dominated age. This proposal is to carry out high level mathematical research in Australia. Basic research on quantum groups underpins applied research and certain areas such as quantum mechanics and string theory. Some structure of quantum groups is too complicated to be seen by even a professional mathematician. A possible interpretation by using representations over a finite field would make it more usable and accessible by computer.Read moreRead less
Modular representations of cyclotomic algebras. This project addresses cutting edge questions in the representation theory of cyclotomic Hecke algebras. Our main focus will be computing decomposition matrices for these algebras. We approach this question from several different directions, each of which will give new insights and lead to significant advances in the theory. The decomposition number problem is important because its' solution gives deep structural information about these algebras wh ....Modular representations of cyclotomic algebras. This project addresses cutting edge questions in the representation theory of cyclotomic Hecke algebras. Our main focus will be computing decomposition matrices for these algebras. We approach this question from several different directions, each of which will give new insights and lead to significant advances in the theory. The decomposition number problem is important because its' solution gives deep structural information about these algebras which can then be applied in other areas. This project will have high impact because cyclotomic Hecke algebras have applications in many different areas and they are currently a hot topic of research in mathematics.Read moreRead less
Representation Theory: Path models and decompositions. The research in this proposal develops tools for capitalising on the benefits of symmetry in large complex systems. These techniques and processes are applicable for solving complex problems in large interactive systems. This project will involve young researchers and train them for problem solving in a wealth of fields, including management, the sciences, the financial industries, and the development of technologies. The research is in o ....Representation Theory: Path models and decompositions. The research in this proposal develops tools for capitalising on the benefits of symmetry in large complex systems. These techniques and processes are applicable for solving complex problems in large interactive systems. This project will involve young researchers and train them for problem solving in a wealth of fields, including management, the sciences, the financial industries, and the development of technologies. The research is in one of the most active cutting edge areas of pure mathematics and will contribute to maintaining Australia's position as a leading nationality in research in representation theory and its applications.
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Invariant theory, cellularity and geometry. Mathematics underpins every aspect of people's interactions with nature (e.g. physics) and with each other (e.g. finance). Its uses range from formulating physical laws in order to understand and predict nature, to analysis of financial concepts and transactions. This project will make fundamental contributions to the mathematics of symmetry. Benefits include enhancement of Australia's position at the very frontier of world class mathematical research, ....Invariant theory, cellularity and geometry. Mathematics underpins every aspect of people's interactions with nature (e.g. physics) and with each other (e.g. finance). Its uses range from formulating physical laws in order to understand and predict nature, to analysis of financial concepts and transactions. This project will make fundamental contributions to the mathematics of symmetry. Benefits include enhancement of Australia's position at the very frontier of world class mathematical research, and a myriad of potential applications to physics, coding theory, information technology, electronic security and experimental design.Read moreRead less