Quantum Spectra. Fundamental quantum processes will play a key role in emerging technologies in the twenty-first century across diverse industries including quantum information technology, quantum computers and electronics, quantum optics, nanoscale quantum microscopes and superconductor technology. Australia has a strong base of expertise in the underpinning quantum disciplines. This project in strategic basic research within mathematical physics will develop a comprehensive and consistent math ....Quantum Spectra. Fundamental quantum processes will play a key role in emerging technologies in the twenty-first century across diverse industries including quantum information technology, quantum computers and electronics, quantum optics, nanoscale quantum microscopes and superconductor technology. Australia has a strong base of expertise in the underpinning quantum disciplines. This project in strategic basic research within mathematical physics will develop a comprehensive and consistent mathematical description of quantum processes. This research will lead to a deeper understanding of quantum processes, keep Australia at the leading edge of international developments and increase Australia's capacity to develop and implement these new technologies.Read moreRead less
Green functions, correlation functions and differential equations. Classical and quantum exact solutions are established cornerstones in Australian applied mathematical research. In this project, we will:- 1). Address long standing open problems, whose resolution will add to mathematical knowledge and enhance Australia's reputation as a leading contributor to these topics; 2). List concrete and tractable sub-projects that will engage young scientists, whose training we are particularly keen on, ....Green functions, correlation functions and differential equations. Classical and quantum exact solutions are established cornerstones in Australian applied mathematical research. In this project, we will:- 1). Address long standing open problems, whose resolution will add to mathematical knowledge and enhance Australia's reputation as a leading contributor to these topics; 2). List concrete and tractable sub-projects that will engage young scientists, whose training we are particularly keen on, in vigorous and internationally competitive research; 3). Facilitate collaborations between various Australian research groups, all of whom are very well positioned to contribute to it; 4). Bring leading scientists to visit Australia to the benefit of the entire Australian mathematical community.Read moreRead less
Three-dimensional geometry and topology. This project will carry out important fundamental research into the geometry and topology of 3-dimensional manifolds, an area of intense activity over the last 30 years.
The work has direct applications to physics, for example recent work in cosmology aimed at determining the global structure of our universe. Our work on knotting and symmetries of molecular graphs will also be of considerable interest in chemistry and biology.
The project will also ....Three-dimensional geometry and topology. This project will carry out important fundamental research into the geometry and topology of 3-dimensional manifolds, an area of intense activity over the last 30 years.
The work has direct applications to physics, for example recent work in cosmology aimed at determining the global structure of our universe. Our work on knotting and symmetries of molecular graphs will also be of considerable interest in chemistry and biology.
The project will also provide high quality training of undergraduate and graduate students in geometry and topology, and will increase international cooperation by developing closer links with colleagues and institutions overseas.Read moreRead less
Rigidity in measured group theory and geometric group theory. Elite universities throughout the world have all made a point of being leaders in the field of pure mathematics. Geometric group theory and orbit equivalence are currently topical areas which attract many of the best young pure mathematicians as is demonstrated by recent invited talks at the International Congress of Mathematicians. This project will foster the development of these fields in Australia as well as nurturing existing e ....Rigidity in measured group theory and geometric group theory. Elite universities throughout the world have all made a point of being leaders in the field of pure mathematics. Geometric group theory and orbit equivalence are currently topical areas which attract many of the best young pure mathematicians as is demonstrated by recent invited talks at the International Congress of Mathematicians. This project will foster the development of these fields in Australia as well as nurturing existing efforts and international links. This proposal will also provide training and research experience for Australian honours and graduate students in mathematics.Read moreRead less
Permanents, permutations and polynomials. The benefits to Australia of fundamental research in core disciplines such as mathematics are well documented. Discrete mathematics and combinatorics are boom disciplines of the computer age and this project seeks new knowledge concerning basic building blocks of combinatorial mathematics. The outcomes will be of interest to theoreticians around the world, enhancing Australia's already high research profile in this crucial area. Importantly, the project ....Permanents, permutations and polynomials. The benefits to Australia of fundamental research in core disciplines such as mathematics are well documented. Discrete mathematics and combinatorics are boom disciplines of the computer age and this project seeks new knowledge concerning basic building blocks of combinatorial mathematics. The outcomes will be of interest to theoreticians around the world, enhancing Australia's already high research profile in this crucial area. Importantly, the project also offers substantial postgraduate training in mathematics, an area in which Australia has an identified skill shortage.Read moreRead less
Generalized group characters. With its numerous international visitors, Ganter's program will be a significant gain for Melbourne as a centre of science and research. Students at all levels will benefit from training in a scientific environment of world-rank.
Ganter plans to build a research community in her field, involving individuals across the nation; this will be a good addition to the Australian research landscape. Finally, she hopes that her past experience in working with minority stud ....Generalized group characters. With its numerous international visitors, Ganter's program will be a significant gain for Melbourne as a centre of science and research. Students at all levels will benefit from training in a scientific environment of world-rank.
Ganter plans to build a research community in her field, involving individuals across the nation; this will be a good addition to the Australian research landscape. Finally, she hopes that her past experience in working with minority students will enable her to contribute to creating a more diverse research community in pure mathematics in Melbourne and across Australia.Read moreRead less
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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The arithmetic of supersingular elliptic curves. The proposed research will have substantial benefits both in the area of pure mathematics, and to the standing of number theory within Australia generally. If successful, the investigators envisage: - fundamental advances in the study of both elliptic curves and modular forms; - key progress in our understanding of the final Millenium Prize Problem in Mathematics; - academic software to compute special values of L-functions; - applications to com ....The arithmetic of supersingular elliptic curves. The proposed research will have substantial benefits both in the area of pure mathematics, and to the standing of number theory within Australia generally. If successful, the investigators envisage: - fundamental advances in the study of both elliptic curves and modular forms; - key progress in our understanding of the final Millenium Prize Problem in Mathematics; - academic software to compute special values of L-functions; - applications to computational mathematics, particularly elliptic curve cryptosystems; - a huge boost to the development of number theory Australia-wide.
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Algorithms and computation in four-dimensional topology. This project will establish Australia as a world leader in computational topology, particularly in the all-important areas of topology in three and four dimensions. In four dimensions this work will be truly groundbreaking; until now the field has seen little development due to the complexity of the algorithms and computations required, and the applicant is in the unique position of having the necessary tools to make significant progress ....Algorithms and computation in four-dimensional topology. This project will establish Australia as a world leader in computational topology, particularly in the all-important areas of topology in three and four dimensions. In four dimensions this work will be truly groundbreaking; until now the field has seen little development due to the complexity of the algorithms and computations required, and the applicant is in the unique position of having the necessary tools to make significant progress in a feasible time frame. In three dimensions this project will strengthen the distinguished computational topology community in Melbourne, led by pioneers such as Rubinstein, Goodman, Hodgson as well as the applicant himself.Read moreRead less
Macdonald polynomials: Combinatorics and representations. This proposal is part of the aim to build a world class research team in algebraic combinatorics and combinatorial representation theory at the University of Melbourne, led by the two CI.
These fields are currently experiencing very rapid growth and development, and a strong Australia based team will further enhance the country's strong reputation in combinatorics and algebra.
The project will also provide a perfect training ground fo ....Macdonald polynomials: Combinatorics and representations. This proposal is part of the aim to build a world class research team in algebraic combinatorics and combinatorial representation theory at the University of Melbourne, led by the two CI.
These fields are currently experiencing very rapid growth and development, and a strong Australia based team will further enhance the country's strong reputation in combinatorics and algebra.
The project will also provide a perfect training ground for Higher Degree Students with interests in pure mathematics as well as computer
algebra and symbolic computation.Read moreRead less