Australian Laureate Fellowships - Grant ID: FL230100088
Funder
Australian Research Council
Funding Amount
$2,531,590.00
Summary
Breakthrough mathematics for dynamical systems and data. This fellowship aims to create a step change in the mathematics we use to learn actionable information from dynamical systems and dynamical data. Using a groundbreaking, operator-theoretic approach to analyse high dimensional systems and spatiotemporal data, this project expects to generate new knowledge in the modelling of complex systems and new pathways for unsupervised machine learning. Expected outcomes of this fellowship include a tr ....Breakthrough mathematics for dynamical systems and data. This fellowship aims to create a step change in the mathematics we use to learn actionable information from dynamical systems and dynamical data. Using a groundbreaking, operator-theoretic approach to analyse high dimensional systems and spatiotemporal data, this project expects to generate new knowledge in the modelling of complex systems and new pathways for unsupervised machine learning. Expected outcomes of this fellowship include a tranche of new mathematics and practical next-generation algorithms to discover hidden human-understandable patterns in complex dynamical systems and data. This should provide significant universal benefits to many areas of science, including elucidating mechanisms underlying climate and social dynamics.Read moreRead less
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
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
Computational modelling of nanofluids for industrial applications. The use of nanoparticles in heat transfer fluids, then known as nanofluids, increases their specific heat and thermal conductivity. Recent experimental works highlight that anomalous transport phenomena are evident in nanofluids that cannot be adequately described by classical conservation laws. We will extend these conservation laws to incorporate fractional operators to capture the fluid memory effects and the impact of particl ....Computational modelling of nanofluids for industrial applications. The use of nanoparticles in heat transfer fluids, then known as nanofluids, increases their specific heat and thermal conductivity. Recent experimental works highlight that anomalous transport phenomena are evident in nanofluids that cannot be adequately described by classical conservation laws. We will extend these conservation laws to incorporate fractional operators to capture the fluid memory effects and the impact of particle clustering. Computational modelling and experimental investigations will be undertaken to identify the heat transfer mechanisms of various nanofluids. The outcomes of the work will increase knowledge on nanofluids and offer a significant opportunity to improve the efficiency of many thermal engineering systems.Read moreRead less
Computer-aided proofs for non-hyperbolic dynamics and blenders . This project aims to develop methods to rigorously detect certain geometric structures in systems that are known to imply chaos and are robust under perturbation. Such structures include blenders and robust heterodimensional cycles and homoclinic tangencies.
This project expects to generate new knowledge in the area of non hyperbolic dynamics utilising a novel combination of recent developments in Dynamical Systems and techniques ....Computer-aided proofs for non-hyperbolic dynamics and blenders . This project aims to develop methods to rigorously detect certain geometric structures in systems that are known to imply chaos and are robust under perturbation. Such structures include blenders and robust heterodimensional cycles and homoclinic tangencies.
This project expects to generate new knowledge in the area of non hyperbolic dynamics utilising a novel combination of recent developments in Dynamical Systems and techniques from rigorous numerics.
Expected outcomes of this project include an efficient computation platform aimed at detecting and verifying chaos-inducing objects in complex dynamical systems.
This should provide significant benefits, such as an increased understanding of non-hyperbolic dynamical systems. Read moreRead less
TSuNAMi: Time Series Network Animal Modelling. Our proposal is motivated by and based upon the successful representation of time series as a network (or graph). We construct an abstract representation of a system from measurements of its changing behaviour over time. Properties of that structure (the network) then allow us to infer diagnostic information of the system. Specifically, we propose to apply this to livestock welfare during transport. By measuring the biological and environment condi ....TSuNAMi: Time Series Network Animal Modelling. Our proposal is motivated by and based upon the successful representation of time series as a network (or graph). We construct an abstract representation of a system from measurements of its changing behaviour over time. Properties of that structure (the network) then allow us to infer diagnostic information of the system. Specifically, we propose to apply this to livestock welfare during transport. By measuring the biological and environment condition of the animal we construct a network representation of that system. Geometric features of that network can then be used to infer health or duress of the subject. This proposal will develop the generic mathematical machinery to connect geometric features of the network with system behaviour. Read moreRead less
New mathematics for understanding complex patterns in the natural sciences. This project aims to examine the interaction of fundamental two-dimensional patterns such as spots and stripes in reaction-diffusion equations, by developing and extending mathematical techniques. These fundamental planar structures form the backbone of more complex patterns and are, for example, observed in models that describe the propagation of impulses in nerve axons and the formation of vegetation patterns. The futu ....New mathematics for understanding complex patterns in the natural sciences. This project aims to examine the interaction of fundamental two-dimensional patterns such as spots and stripes in reaction-diffusion equations, by developing and extending mathematical techniques. These fundamental planar structures form the backbone of more complex patterns and are, for example, observed in models that describe the propagation of impulses in nerve axons and the formation of vegetation patterns. The future impact of this research will have economic and environmental benefits. For example, the project will develop a deeper understanding of interacting patterns that will provide insights into the role of vegetation in ecosystems that are undergoing desertification.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
A coordinate-independent theory for multi-time-scale dynamical systems. Biochemical reaction networks operate inherently on many disparate timescales, and identifying this temporal hierarchy is key to understanding biological behaviour. Currently, the existing dynamical systems theory is not able to rigorously analyse many important biological systems and networks due to this inherent non-standard multi-time-scale splitting. This project aims to remove these stumbling blocks and develop a coordi ....A coordinate-independent theory for multi-time-scale dynamical systems. Biochemical reaction networks operate inherently on many disparate timescales, and identifying this temporal hierarchy is key to understanding biological behaviour. Currently, the existing dynamical systems theory is not able to rigorously analyse many important biological systems and networks due to this inherent non-standard multi-time-scale splitting. This project aims to remove these stumbling blocks and develop a coordinate-independent mathematical theory that weaves together results from geometric singular perturbation theory, differential and algebraic geometry and reaction network theory to decompose and explain the structure in the dynamic hierarchy of events in non-standard multi-time-scale systems and networks.Read moreRead less