Pictures for Operator Algebras: higher rank graphs. The runaway success of graph C*-algebras pioneered at the University of Newcastle places Australia at the forefront of an exciting new research field. Higher-dimensional graphs and their algebras are a promising new direction, and this project will keep Australian researchers at its cutting edge.
This project will involve and train talented young researchers who will contribute to the Mathematical outcomes of the current project. Their invol ....Pictures for Operator Algebras: higher rank graphs. The runaway success of graph C*-algebras pioneered at the University of Newcastle places Australia at the forefront of an exciting new research field. Higher-dimensional graphs and their algebras are a promising new direction, and this project will keep Australian researchers at its cutting edge.
This project will involve and train talented young researchers who will contribute to the Mathematical outcomes of the current project. Their involvement will train them in the techniques of modern Mathematics and furnish them with important international research connections. In doing so we shall be laying a strong foundation for the future of Australian Mathematical research.Read moreRead less
Noncommutative geometry of groups acting on buildings. Consider a tiling of the plane by triangles, where each triangle is labeled by an element of a finite alphabet. Suppose that only certain pairs of labels are allowed to be adjacent to each other in each direction. The tiled planes can be pasted together to form the abstract mathematical object known as a building. This building and its boundary, give rise to new families of C*-algebras and groups. The algebras have a rich structure which it ....Noncommutative geometry of groups acting on buildings. Consider a tiling of the plane by triangles, where each triangle is labeled by an element of a finite alphabet. Suppose that only certain pairs of labels are allowed to be adjacent to each other in each direction. The tiled planes can be pasted together to form the abstract mathematical object known as a building. This building and its boundary, give rise to new families of C*-algebras and groups. The algebras have a rich structure which it is proposed to investigate and link with geometric properties of the groups. New insights into geometry, dynamics and algebra are expected.Read moreRead less
Operator algebras associated to product systems, and higher-rank-graph algebras. Operator algebras are used to study a wide range of physical systems in quantum physics and quantum computing, and in electrical engineering. The clearer our picture of how operator algebras work, the better we are able to predict and explain how these physical systems will behave. The proposed research project is aimed at showing that we can describe operator algebras in terms of simple coloured diagrams rather tha ....Operator algebras associated to product systems, and higher-rank-graph algebras. Operator algebras are used to study a wide range of physical systems in quantum physics and quantum computing, and in electrical engineering. The clearer our picture of how operator algebras work, the better we are able to predict and explain how these physical systems will behave. The proposed research project is aimed at showing that we can describe operator algebras in terms of simple coloured diagrams rather than abstract mathematical symbols. Consequently, the project will lead to a simpler and less technical approach to the physical problems which operator algebras are used to study.Read moreRead less
Modular Index Theory. This project capitilises on Australian advances in mathematics, particularly noncommutative geometry. It will maintain and extend Australia's prominence in this subject, providing excellent opportunities for young researchers via the research networks this project will establish. Being at the interface of ideas in mathematics and physics, there is potential for future technological spin offs for Australia.