Totally disconnected groups in algebra and geometry. Mathematics research creates and develops new concepts for understanding the world. Group theory is a branch of mathematics based on our innate sense 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 made significant breakthroughs in the study of sym ....Totally disconnected groups in algebra and geometry. Mathematics research creates and develops new concepts for understanding the world. Group theory is a branch of mathematics based on our innate sense 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 made significant breakthroughs in the study of symmetry groups of networks, giving Australia an international lead in this research area. The project will develop the insights gained to make Australia a centre of expertise on these symmetry groups, which have applications to many areas including information and communication technology.Read moreRead less
Applications of Group Theory to Finite Geometry. Group theory and geometry have influenced one another for over a century. The most important structures in geometry are the symmetric ones and the most important groups act on geometries. Recent developments in finite geometry, although informed by symmetry, have used a minimum of group theory. The project aims to redress this, by applying results from a broad range of finite group theory to the presently hot topics in finite geometry. Our aim is ....Applications of Group Theory to Finite Geometry. Group theory and geometry have influenced one another for over a century. The most important structures in geometry are the symmetric ones and the most important groups act on geometries. Recent developments in finite geometry, although informed by symmetry, have used a minimum of group theory. The project aims to redress this, by applying results from a broad range of finite group theory to the presently hot topics in finite geometry. Our aim is to achieve a paradigm shift, by finding substantively different structures than those presently known. Should it succeed, much activity in geometry would follow, seeking geometric interpretation of these group theoretic results. Our focus is necessitated by the lack of a result characterising the underlying groups of symmetric generalised quadrangles.Read moreRead less
Permutation groups and their interplay with symmetry in finite geometry and graph theory. A strong mathematical community in Australia provides the foundations for future discoveries in technology, science and business. The use of group theory to characterise symmetric generalised quadrangles, partial quadrangles, and strongly regular graphs, and the construction of new examples of such objects, will enhance Australia's leading position in Group Theory, Algebraic Graph Theory and Finite Geometry ....Permutation groups and their interplay with symmetry in finite geometry and graph theory. A strong mathematical community in Australia provides the foundations for future discoveries in technology, science and business. The use of group theory to characterise symmetric generalised quadrangles, partial quadrangles, and strongly regular graphs, and the construction of new examples of such objects, will enhance Australia's leading position in Group Theory, Algebraic Graph Theory and Finite Geometry. This project will also strengthen the collaboration between Australian, Belgian and Italian Universities and support young researchers, developing expertise in a world-leading research group, to drive Australia's future in mathematics.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
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
Braid monoids, presentations and normal forms. Braid groups arise naturally in various areas of mathematics, physics and computer science including knot theory, Lie theory, quantum groups and cryptography. There is a uniform geometric description of braid groups; however this is not the case algebraically. This project aims to find the connections between the algebra, combinatorics and geometry of braid groups in order to obtain a uniform algebraic description. This generalisation will allow adv ....Braid monoids, presentations and normal forms. Braid groups arise naturally in various areas of mathematics, physics and computer science including knot theory, Lie theory, quantum groups and cryptography. There is a uniform geometric description of braid groups; however this is not the case algebraically. This project aims to find the connections between the algebra, combinatorics and geometry of braid groups in order to obtain a uniform algebraic description. This generalisation will allow advances in the related areas of mathematics and physics. In addition to theoretical results, new algorithms for calculating in braid groups will be given, which can then be implemented computationally.
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Geometry on Nilpotent Groups. Nilpotent Lie groups turn up in mechanics, robotics, biology, physical chemistry and electrical engineering, to deal with real-world configurations in which it is not possible to move in all directions. This project will develop the mathematical foundations of the theory in order to underpin the many and varied applications. Development of the foundations also allows techniques developed to deal with one application to be transferred to deal with another applicati ....Geometry on Nilpotent Groups. Nilpotent Lie groups turn up in mechanics, robotics, biology, physical chemistry and electrical engineering, to deal with real-world configurations in which it is not possible to move in all directions. This project will develop the mathematical foundations of the theory in order to underpin the many and varied applications. Development of the foundations also allows techniques developed to deal with one application to be transferred to deal with another application. The project will also raise the profile of Australian Mathematics internationally and train the researchers of the future.Read moreRead less
Geometric Group Theory. Groups arise naturally as symmetries of geometric objects. Often groups have an interesting geometric structure obtained by thinking of these geometric objects coursely. This project aims to study the subgroup structure of such groups and obtain homological, geometric and algorithmic information. It further investigates natural decompositions of groups with geometric structure along special subgroups so that the factors have simpler properties.{P
Affine buildings and Hecke algebras. This project is breakthrough science. Affine buildings, Ramanujan complexes and the representation theory of affine Hecke algebras are on the cutting edge of research in mathematics, as evidenced by recent special programs at Cambridge and Princeton. The outcomes from this project will be published in first class journals and they will be implemented in computer algebra systems for world wide application. The project is likely to have flow-on effects in othe ....Affine buildings and Hecke algebras. This project is breakthrough science. Affine buildings, Ramanujan complexes and the representation theory of affine Hecke algebras are on the cutting edge of research in mathematics, as evidenced by recent special programs at Cambridge and Princeton. The outcomes from this project will be published in first class journals and they will be implemented in computer algebra systems for world wide application. The project is likely to have flow-on effects in other disciplines, notably communication networks, mathematical physics and computer science.
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