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
Totally disconnected groups, representations and discrete mathematics. This project involves participation in programs at the Institute of Advanced Studies in Princeton and the nearby Center for Discrete Mathematics and Theoretical Computer Science that are designed to initiate collaborations across distinct mathematical research areas. These programs will set future research directions and could lead to innovations in computer science. Discoveries I have made in one of the research areas mean ....Totally disconnected groups, representations and discrete mathematics. This project involves participation in programs at the Institute of Advanced Studies in Princeton and the nearby Center for Discrete Mathematics and Theoretical Computer Science that are designed to initiate collaborations across distinct mathematical research areas. These programs will set future research directions and could lead to innovations in computer science. Discoveries I have made in one of the research areas mean that I may be able to make substantial contributions to these programs. Early involvement in influential programs such as these means that Australia is well placed to take advantage of developments that result and also enhances the reputation of Australian mathematics.Read moreRead less
Geometric representation of small-rank totally disconnected groups. Mathematics research creates and develops new concepts for understanding the world. Group theory is a branch of mathematics based on our innate sense of of symmetry. It was invented 200 hundred years ago and has grown into a language for analysing and classifying things ranging from wallpaper patterns to crystals, the fundamental particles of physics and Rubik's cube. The chief investigators have significant breakthroughs in the ....Geometric representation of small-rank totally disconnected groups. Mathematics research creates and develops new concepts for understanding the world. Group theory is a branch of mathematics based on our innate sense of of symmetry. It was invented 200 hundred years ago and has grown into a language for analysing and classifying things ranging from wallpaper patterns to crystals, the fundamental particles of physics and Rubik's cube. The chief investigators have significant breakthroughs in the study of symmetry groups of networks, giving Australia an international lead in this research. The project will develop the insights gained to make Australia a centre of expertise on these symmetry groups, which have applications to information and communication technology, among many others.Read moreRead less
Reflection groups. The study of symmetry in geometrical and abstract contexts is a central issue in such diverse areas as mathematical physics, singularity theory, algebraic geometry, quantum groups and the study of knots and braids. Group theory provides the mathematical framework for the analysis of symmetry. Reflection groups, simple examples of which are the symmetry groups of the five platonic solids, play a key role in all of the areas mentioned above. Thus an improved understanding of ref ....Reflection groups. The study of symmetry in geometrical and abstract contexts is a central issue in such diverse areas as mathematical physics, singularity theory, algebraic geometry, quantum groups and the study of knots and braids. Group theory provides the mathematical framework for the analysis of symmetry. Reflection groups, simple examples of which are the symmetry groups of the five platonic solids, play a key role in all of the areas mentioned above. Thus an improved understanding of reflection groups will significantly enhance the development of several important theories.
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Quantized representation theory. The representation theory of quantized algebras, or deformation algebras, is a rapidly expanding and exciting field. It has wide ranging applications from within mathematics, to knot theory and statistical mechanics. This project addresses several important open problems in the area with an emphasis on structural innovations and computing explicit numerical invariants.
Continued Fractions and Torsion on Hyperelliptic Curves. Scientific advance should not blindly add to our knowledge; a true advance brings insights that collapse different issues into one. Understanding more is to need to remember less. For an important class of examples, this project identifies the study of a fundamental invariant of a quadratic number field, its regulator and hence its class number, with maximum torsion on the Jacobian variety of an hyperelliptic curve. The investigator's meth ....Continued Fractions and Torsion on Hyperelliptic Curves. Scientific advance should not blindly add to our knowledge; a true advance brings insights that collapse different issues into one. Understanding more is to need to remember less. For an important class of examples, this project identifies the study of a fundamental invariant of a quadratic number field, its regulator and hence its class number, with maximum torsion on the Jacobian variety of an hyperelliptic curve. The investigator's methods will surprise some longstanding problems into submission and in particular will lead them to reveal full data on torsion on hyperelliptic curves of low genus.
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Improving Upper and Lower Bounds on the Order of Large Graphs under Degree and Distance Constraints. Networks govern all aspects of society, including transportation networks, communication networks, computer networks and networks for the distribution of goods etc. - and the theoretical analysis of such networks has become a subject of fundamental importance. Networks can be modelled by graphs. This project will provide new theoretical results which will improve our knowledge of network topologi ....Improving Upper and Lower Bounds on the Order of Large Graphs under Degree and Distance Constraints. Networks govern all aspects of society, including transportation networks, communication networks, computer networks and networks for the distribution of goods etc. - and the theoretical analysis of such networks has become a subject of fundamental importance. Networks can be modelled by graphs. This project will provide new theoretical results which will improve our knowledge of network topologies. The new knowledge will then be utilised in the construction of large graphs with respect to given maximum degree and distance constraints.Read moreRead less