A novel framework for designing input excitation for system identification. Engineers need mathematical models describing the behaviour of the components they use in their design. This project aims at resolving some critical issues faced by the researchers developing cutting edge mathematical software for building such models.
What predictions can I trust? Stability of chaotic random dynamical systems. This project aims to make significant progress on the intricate question of global stability of non-autonomous chaotic dynamical systems. Using ergodic theory, this project expects to determine when and how errors in dynamical models that are small and frequent, or large and infrequent, can cause dramatic changes in meaningful mathematical model outputs. Expected outcomes include the discovery of mathematical mechanisms ....What predictions can I trust? Stability of chaotic random dynamical systems. This project aims to make significant progress on the intricate question of global stability of non-autonomous chaotic dynamical systems. Using ergodic theory, this project expects to determine when and how errors in dynamical models that are small and frequent, or large and infrequent, can cause dramatic changes in meaningful mathematical model outputs. Expected outcomes include the discovery of mathematical mechanisms underlying large-scale (in)stability for time-dependent dynamical systems, and reliable numerical methods for detecting instabilities. This research is expected to lead to improved characterisations of shocks or collapse in externally driven dynamical systems and assist scientists to gauge which predictions they can trust.Read moreRead less
Modern mathematics to unravel the birth of coherence in dynamical systems. This project aims to reveal the precise mathematical mechanisms underlying the emergence and disappearance of long-lived coherent features in dynamical systems. This project expects to generate new fundamental mathematics in the area of dynamical systems, using innovative operator-theoretic approaches to carefully tease apart the lifecycles of coherent structures. The expected outcomes of this project include new mathemat ....Modern mathematics to unravel the birth of coherence in dynamical systems. This project aims to reveal the precise mathematical mechanisms underlying the emergence and disappearance of long-lived coherent features in dynamical systems. This project expects to generate new fundamental mathematics in the area of dynamical systems, using innovative operator-theoretic approaches to carefully tease apart the lifecycles of coherent structures. The expected outcomes of this project include new mathematical theory and computational algorithms to anticipate the genesis and destruction of coherent objects, which are key organisers of complex geophysical flows. This breakthrough mathematics should provide significant benefits, such as improved prediction of eddy transport and persistence of weather and climate patterns.Read moreRead less
New mathematics to quantify fluctuations and extremes in dynamical systems. Many problems in the natural world result from the cumulative effect of extreme events in complex dynamical systems. Dynamical models of ecological and physical processes have internal variables that can combine to produce large observable changes. Quantitative estimation of the variability of these chaotic models is difficult because of the time dependence of the dynamics and their “long memory” due to significant deter ....New mathematics to quantify fluctuations and extremes in dynamical systems. Many problems in the natural world result from the cumulative effect of extreme events in complex dynamical systems. Dynamical models of ecological and physical processes have internal variables that can combine to produce large observable changes. Quantitative estimation of the variability of these chaotic models is difficult because of the time dependence of the dynamics and their “long memory” due to significant deterministic components. This project aims to develop mathematics and numerics to accurately quantify and assess these complicated variations. The project expects to provide powerful tools to predict harmful outcomes in biogeophysical systems, and assist with the development of mitigation strategies.Read moreRead less
Mathematical modelling can provide vital information on the effectiveness and practical implementation of microbicides and vaccines against HIV. This project will produce mathematical models of the earliest stages of HIV infection suitable for investigation of the implementation of vaccines and microbicides. It will provide a framework to investigate why these interventions have performed poorly to date, and how these may be better implemented.
Spectral Theory of Hamiltonian Dynamical Systems. Stability theory of steady states, travelling waves, periodic waves, and other coherent structures in nonlinear Hamiltonian partial differential equations is a cornerstone of modern dynamical systems. In particular it is of utmost importance to reliably compute eigenvalues, which determine the stability or instability of such structures. This project will develop methods to compute the spectrum of Hamiltonian operators in more than one spatial di ....Spectral Theory of Hamiltonian Dynamical Systems. Stability theory of steady states, travelling waves, periodic waves, and other coherent structures in nonlinear Hamiltonian partial differential equations is a cornerstone of modern dynamical systems. In particular it is of utmost importance to reliably compute eigenvalues, which determine the stability or instability of such structures. This project will develop methods to compute the spectrum of Hamiltonian operators in more than one spatial dimension. It will use the powerful geometric tools of the Maslov index and the Evans function. We will use these to simultaneously advance, and bring together the theories of the two dimensional Euler equations and Jacobi operators.Read moreRead less
Extracting macroscopic variables and their dynamics in multiscale systems with metastable states. There are practical barriers to the simulation of complex systems such as molecular systems and the climate system because of the high-dimensionality of the models and the presence of multiscale dynamics. This project will lift these barriers by uncovering the most relevant variables and by creating innovative multiscale simulation algorithms.
A geometric theory for non-standard relaxation oscillators. This project aims to develop new geometric methods for the analysis of multi-scale models of biological rhythms, and design diagnostic tools to identify key parameters that cause and control these signals. Rhythms, such as breathing, neural and cardiac rhythms and pulsatile hormone secretion, are central for life. Many important biochemical cell signals exhibiting relaxation-type behaviour cannot be rigorously analysed with standard dy ....A geometric theory for non-standard relaxation oscillators. This project aims to develop new geometric methods for the analysis of multi-scale models of biological rhythms, and design diagnostic tools to identify key parameters that cause and control these signals. Rhythms, such as breathing, neural and cardiac rhythms and pulsatile hormone secretion, are central for life. Many important biochemical cell signals exhibiting relaxation-type behaviour cannot be rigorously analysed with standard dynamical systems tools due to an inherent non-uniform time-scale splitting in these models. This project aims to develop a unified mathematical theory that weaves together results from geometric singular perturbation theory and algebraic geometry to explain the genesis of complex rhythms and patterns in biological, non-standard, multi-scale systems, both at individual and network level.Read moreRead less
Microcantilevers for multifrequency atomic force microscopy. This project aims to design a microcantilever with high-performing sensors more sensitive and with better noise performance than the typical optical system used in commercial Atomic Force Microscopes (AFMs). The AFM, a nanotechnology instrument, uses a microcantilever (with an extremely shape probe) to interrogate a sample surface. It has made important discoveries in nanotechnology, life sciences, nanomachining, material science and d ....Microcantilevers for multifrequency atomic force microscopy. This project aims to design a microcantilever with high-performing sensors more sensitive and with better noise performance than the typical optical system used in commercial Atomic Force Microscopes (AFMs). The AFM, a nanotechnology instrument, uses a microcantilever (with an extremely shape probe) to interrogate a sample surface. It has made important discoveries in nanotechnology, life sciences, nanomachining, material science and data storage systems. Despite its success, the technique’s spatial resolution and quantitative measurements are limited. This project could lead to breakthrough technologies such as atomic force spectroscopy to study elastic modulus of nanostructures, and establish Australia's prominence in this emerging field.Read moreRead less
Atomic Resolution Sensors for Imaging and Metrological Science. This project aims to create new sensing technologies for detecting motion on the atomic scale with Megahertz (MHz) bandwidth. Advanced signal processing and communication theory will be applied with the aim of developing new classes of capacitive, inductive and optical position sensors. The resolution and bandwidth are predicted to be a one-hundred fold improvement over the current state-of-the-art. Applications are expected to incl ....Atomic Resolution Sensors for Imaging and Metrological Science. This project aims to create new sensing technologies for detecting motion on the atomic scale with Megahertz (MHz) bandwidth. Advanced signal processing and communication theory will be applied with the aim of developing new classes of capacitive, inductive and optical position sensors. The resolution and bandwidth are predicted to be a one-hundred fold improvement over the current state-of-the-art. Applications are expected to include biomedical imaging, high-speed nanofabrication, high-resolution computer numerical control (CNC) machining, high-speed gas and chemical sensors, and ultra-precise seismometers and gyroscopes.Read moreRead less