Computer Assisted Research Mathematics and its Applications. The mathematics community will benefit from infusion of new computer-assisted techniques and modalities for research and training post-graduate students, both from my pure research project and through development of an associated research centre. Ultimately, this should also help more school students learn mathematics well and so play a part in addressing Australia's skill shortage. Also, the work on optimization algorithms promises to ....Computer Assisted Research Mathematics and its Applications. The mathematics community will benefit from infusion of new computer-assisted techniques and modalities for research and training post-graduate students, both from my pure research project and through development of an associated research centre. Ultimately, this should also help more school students learn mathematics well and so play a part in addressing Australia's skill shortage. Also, the work on optimization algorithms promises to improve the performance and quality of many practical signal reconstruction methods. These are used by varied Australian industries from telecommunication to mining and by researchers in the digital arts and fields such as astronomy, physics, chemistry, bioscience, geoscience, engineering and medicine.Read moreRead less
Mathematics of Cryptography. The Australian economy and society requires fast, reliable, and secure communication. Current first-generation security solutions are not capable of supporting the efficiency and scalability requirements of mass-market adoption of wireless and embedded consumer applications. New security infrastructures are emerging and must be carefully, but rapidly, defined. Thus developing new mathematically solid tools in this area is an important and urgent tasks. In addition, t ....Mathematics of Cryptography. The Australian economy and society requires fast, reliable, and secure communication. Current first-generation security solutions are not capable of supporting the efficiency and scalability requirements of mass-market adoption of wireless and embedded consumer applications. New security infrastructures are emerging and must be carefully, but rapidly, defined. Thus developing new mathematically solid tools in this area is an important and urgent tasks. In addition, the intended work advances our knowledge of the theory and the quality of our culture. As such, it will promote the Australian science and will also have many practical applications in Cryptography, Computer Security and E-Commerce.Read moreRead less
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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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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Representation theory of groups and applications to geometry and number theory. Representation theory is at the center of the mathematical study of symmetry, which we constantly use to understand the world. Combine with geometry this theory produces spectacular results in number theory. This project aims to study p-adic phenomena in these theories. Its
main outcomes will be p-adic automorphic forms and local functoriality.
High dimensional problems of integration and approximation. In many applications, notably financial mathematics, problems of
integration and approximation of functions in very high dimensions
are of great interest. By finding modern mathematical solutions to
these problems, we will therefore contribute to Australia's future
success in developing innovative technologies for industrial and
economic applications. By researching at an internationally
competitive level and by cooperating with i ....High dimensional problems of integration and approximation. In many applications, notably financial mathematics, problems of
integration and approximation of functions in very high dimensions
are of great interest. By finding modern mathematical solutions to
these problems, we will therefore contribute to Australia's future
success in developing innovative technologies for industrial and
economic applications. By researching at an internationally
competitive level and by cooperating with international experts, we
will have a share in further strengthening the excellent role of
Australian research institutions within the international scientific
community in mathematics and scientific computing.Read moreRead less
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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Mathematics of Cryptography. The Australian society and economy requires fast, reliable, and secure communication. First-generation security solutions are not capable of supporting the efficiency and scalability requirements of mass-market adoption of wireless and embedded consumer applications. New security infrastructures are emerging and must be carefully, but rapidly, defined. Thus developing new mathematically solid tools in this area is one of the most important and urgent tasks. Besides, ....Mathematics of Cryptography. The Australian society and economy requires fast, reliable, and secure communication. First-generation security solutions are not capable of supporting the efficiency and scalability requirements of mass-market adoption of wireless and embedded consumer applications. New security infrastructures are emerging and must be carefully, but rapidly, defined. Thus developing new mathematically solid tools in this area is one of the most important and urgent tasks. Besides, the intended work advances our knowledge of the theory and the quality of our culture. As such, it will promote the Australian science and will also have many practical applications in Cryptography, Computer Security and E-Commerce.Read moreRead less
Number Theoretic Methods in Cryptography. It is well known that Number Theory, besides its intrinsic beauty, provides many powerful tools for modern Cryptography. The aim of the project is to formulate and solve new and important mathematical problems, which lie in the background of modern cryptography. They are also of independent value for pure mathematics because they very often stimulate new approaches to and new surprising points of view on classical results and methods. The main outcome w ....Number Theoretic Methods in Cryptography. It is well known that Number Theory, besides its intrinsic beauty, provides many powerful tools for modern Cryptography. The aim of the project is to formulate and solve new and important mathematical problems, which lie in the background of modern cryptography. They are also of independent value for pure mathematics because they very often stimulate new approaches to and new surprising points of view on classical results and methods. The main outcome will be advancing our theoretical knowledge about several major cryptosystems. The project will extend and enrich the area of applications of mathematics to cryptography and related areas.Read moreRead less