Dynamics of eigenvalue/eigenspace algorithms with applications to signal processing. Many problems in signal and systems lead naturally to an eigenvalue/eigenspace determination and tracking problem; for example (acoustic) echo-cancellation, crosstalk suppression in ADSL modems, direction of arrival determination with an array of sensors, linear system identification etc. Exploiting methods from global analysis and dynamical systems theory we will study the available algorithms for eigenspace de ....Dynamics of eigenvalue/eigenspace algorithms with applications to signal processing. Many problems in signal and systems lead naturally to an eigenvalue/eigenspace determination and tracking problem; for example (acoustic) echo-cancellation, crosstalk suppression in ADSL modems, direction of arrival determination with an array of sensors, linear system identification etc. Exploiting methods from global analysis and dynamical systems theory we will study the available algorithms for eigenspace determination to characterise their computational efficiency, accuracy and effectiveness in various data scenarios. The analysis will lead to improved designs for eigenvalue/eigenspace algorithms, as well as design tools to engineer algorithms to specific situations.Read moreRead less
Special Research Initiatives - Grant ID: SR0354466
Funder
Australian Research Council
Funding Amount
$20,000.00
Summary
Mathematics in Contemporary Science. The Mathematics in Contemporary Science Research Network brings contemporary methods of non-linear analysis and differential equations, geometric reasoning and relevant algebraic and topological ideas to enrich six application areas in modern science: Complex Systems, Computer Vision, Optimal Transportation, Nanotechnology, Physics and Shortest Networks. MiCS will develop both the mathematics and the application areas in parallel. It will focus on postgradu ....Mathematics in Contemporary Science. The Mathematics in Contemporary Science Research Network brings contemporary methods of non-linear analysis and differential equations, geometric reasoning and relevant algebraic and topological ideas to enrich six application areas in modern science: Complex Systems, Computer Vision, Optimal Transportation, Nanotechnology, Physics and Shortest Networks. MiCS will develop both the mathematics and the application areas in parallel. It will focus on postgraduate training through workshops, summer schools and web based resources and build long-term international collaborations with EU networks and NSERC, NSF and EPSRC institutes as well as bringing together academic and industry leaders.Read moreRead less
Higher dimensional methods for algebras and dynamical systems. Australian researchers have pioneered recent research in combinatorial C*-algebras. We are now uniquely placed to capitalise on this situation to make significant connections with research in dynamical systems. This project will thus position Australian mathematics at the nexus of two important research areas.
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
Harmonic analysis and dispersive partial differential equations. This project aims to develop theoretical results and practical techniques in the study of Partial Differential Equations. Harmonic analysis is used to study these equations; in which a system’s local behaviour is used to analyse global properties, using techniques such as the Fourier transform. The project will investigate central problems in the area, revealing deep connections between analysis and geometry, and apply these to stu ....Harmonic analysis and dispersive partial differential equations. This project aims to develop theoretical results and practical techniques in the study of Partial Differential Equations. Harmonic analysis is used to study these equations; in which a system’s local behaviour is used to analyse global properties, using techniques such as the Fourier transform. The project will investigate central problems in the area, revealing deep connections between analysis and geometry, and apply these to study the solutions’ long-term behaviour to non-linear equations. Expected outcomes include theoretical results and practical techniques to solve non-linear dispersive equations, which arise in quantum and fluid mechanics.Read moreRead less
Curvature flows and spectral estimates. Curvature flows are a class of geometrically motivated equations, modelled on the heat equation. Recently, researchers have developed new methods for studying the regularity of solutions to these equations, and applied them to a different problem, that of estimating quantities depending on the smaller eigenvalues of a Schroedinger operator. This project builds on the early success of this research and will produce a new understanding of the behaviour of ei ....Curvature flows and spectral estimates. Curvature flows are a class of geometrically motivated equations, modelled on the heat equation. Recently, researchers have developed new methods for studying the regularity of solutions to these equations, and applied them to a different problem, that of estimating quantities depending on the smaller eigenvalues of a Schroedinger operator. This project builds on the early success of this research and will produce a new understanding of the behaviour of eigenvalues, establish sharp estimates for spectral quantities, particularly on manifolds with curvature bounds, and find optimal conditions under which non-compact solutions to curvature flows are stable.Read moreRead less
Coset spaces and Hecke algebra actions. This project will develop fundamental models and their mechanics as tools for studying subtle geometry and hidden symmetry in number systems and systems of equations. These powerful new models will provide an elementary and tractable approach for exploiting patterns that are naturally embedded in complex systems.
Geometric methods in representation theory and the Langlands program. This research project aims to study questions in representation theory of groups using geometric methods. A central role is played by Langlands program which, broadly understood, can be viewed as a grand unified theory of mathematics. One setting for the work is modular representation theory with the aim of understanding irreducible characters. The project also aims to work on combinatorics and geometry in algebraic groups in ....Geometric methods in representation theory and the Langlands program. This research project aims to study questions in representation theory of groups using geometric methods. A central role is played by Langlands program which, broadly understood, can be viewed as a grand unified theory of mathematics. One setting for the work is modular representation theory with the aim of understanding irreducible characters. The project also aims to work on combinatorics and geometry in algebraic groups in small characteristics and one goal is to obtain a more uniform geometric understanding across all characteristics. The project also aims to work in the context of real groups and with the Gukov-Witten "fix of the orbit method" via branes. Finally, the project expects to begin a study of deformations of Galois representations in a general context.Read moreRead less
Tantalizer algebras and generalized lattice models. This project exploits underlying symmetry to characterise components and flow patterns in network configurations. The project will develop tools for analysis and optimisation of systems of interacting nodes which can arise in materials, computing networks, and any social or industrial contexts with communication or product transfer between nodes.
Theory and applications of symmetries of relational structures. Relational structures underpin information science, and symmetries of the structures determine their behaviour. This project exploits the interplay between several distinct branches of mathematics concerned with symmetries of relational structures, and will lead to new understandings and breakthroughs of high theoretical interest.