Discovery Early Career Researcher Award - Grant ID: DE210101352
Funder
Australian Research Council
Funding Amount
$330,000.00
Summary
Inverting the Signature Transform for Rough Paths and Random Processes. The signature transform provides an effective summary of the essential information encoded in multidimensional paths that are highly oscillatory and involve complicated randomness. The main goal of this project is to develop new algorithmic methods to reconstruct rough paths and random processes from the signature transform at various quantitative levels. This project expects to make theoretical breakthrough on the significa ....Inverting the Signature Transform for Rough Paths and Random Processes. The signature transform provides an effective summary of the essential information encoded in multidimensional paths that are highly oscillatory and involve complicated randomness. The main goal of this project is to develop new algorithmic methods to reconstruct rough paths and random processes from the signature transform at various quantitative levels. This project expects to make theoretical breakthrough on the significant open problem of signature inversion, thereby advancing knowledge in the areas of rough path theory and stochastic analysis. The newly developed methods will be utilised in combination with the emerging signature-based approach to study important problems in financial data analysis and visual speech recognition.
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Singular solutions for nonlinear elliptic and parabolic equations. The analysis of many models fundamental to physical and biological sciences is obstructed by singularities. This project aims to discover and classify the singular solutions for two important types of nonlinear equations: elliptic and parabolic. The project expects to generate novel methods to decipher singularities by using innovative approaches from geometric analysis and dynamical systems. Expected outcomes of this project inc ....Singular solutions for nonlinear elliptic and parabolic equations. The analysis of many models fundamental to physical and biological sciences is obstructed by singularities. This project aims to discover and classify the singular solutions for two important types of nonlinear equations: elliptic and parabolic. The project expects to generate novel methods to decipher singularities by using innovative approaches from geometric analysis and dynamical systems. Expected outcomes of this project include new and powerful tools to advance a more general theory of singularities. This should provide significant benefits, such as new mathematical knowledge on key issues on singularities lying at the forefront of international research and enhanced expertise in an area of worldwide recognition for Australia.Read moreRead less
Nonlinear partial differential equations with anisotropy and singularities. This project aims to develop new methods in the study of several classes of nonlinear partial differential equations featuring singularities and nonstandard growth conditions. The understanding of countless phenomena in physical and biological sciences is impaired by singularities arising naturally in the models of nonlinear partial differential equations. In a systematic study of singularities on important problems, thi ....Nonlinear partial differential equations with anisotropy and singularities. This project aims to develop new methods in the study of several classes of nonlinear partial differential equations featuring singularities and nonstandard growth conditions. The understanding of countless phenomena in physical and biological sciences is impaired by singularities arising naturally in the models of nonlinear partial differential equations. In a systematic study of singularities on important problems, this project aims to advance new analytical methods and settle fundamental questions that remain open. Outcomes include a more inclusive singularity theory, which fully describes all the singularities that can occur. More immediate applications are in core areas of mathematics, which bear significance to quantum mechanics and image processing in particular.Read moreRead less
The shape of chaos: geometric advances in partially hyperbolic dynamics. This project aims to use recent advances in geometry and topology to discover new forms of chaotic dynamical systems and further classify the forms of chaos which are possible. Many systems in nature exhibit chaotic dynamics as they change in time. Not all systems are chaotic in the same way, and identifying the form of chaos and its qualitative properties is crucial to truly understanding the system. Chaotic dynamics aris ....The shape of chaos: geometric advances in partially hyperbolic dynamics. This project aims to use recent advances in geometry and topology to discover new forms of chaotic dynamical systems and further classify the forms of chaos which are possible. Many systems in nature exhibit chaotic dynamics as they change in time. Not all systems are chaotic in the same way, and identifying the form of chaos and its qualitative properties is crucial to truly understanding the system. Chaotic dynamics arise in chemical reactions, celestial mechanics, industrial mixing processes, fusion reactors, and many other processes. This project will aid in predicting the possible long-term behaviours of these systems.Read moreRead less
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
Interplay between Ergodic Theory, Additive Combinatorics and Ramsey Theory. This project aims to address fundamental problems in Number Theory and Combinatorics by developing new innovative ergodic theoretic methods. Expected outcomes of the project include finding new patterns in dense subsets of trees, obtaining rigorous number-theoretic results emphasising the independence of addition and multiplication, finding infinite patterns in dense subsets of primes, and developing a multi-dimensional ....Interplay between Ergodic Theory, Additive Combinatorics and Ramsey Theory. This project aims to address fundamental problems in Number Theory and Combinatorics by developing new innovative ergodic theoretic methods. Expected outcomes of the project include finding new patterns in dense subsets of trees, obtaining rigorous number-theoretic results emphasising the independence of addition and multiplication, finding infinite patterns in dense subsets of primes, and developing a multi-dimensional analogue of the dense model theory for primes. This project will provide significant benefits to Australian research via an intensive collaboration with best international and Australian researchers working in ergodic and number theory as well as will be used to educate a new generation of Australian students. 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
Additive combinatorics of infinite sets via ergodic theoretic approach. The proposed project will utilise innovative ergodic theoretic approaches to enable us to address important questions in Additive Combinatorics (Number Theory) and Fractal Geometry. In particular, we will resolve long-standing inverse additive problems for infinite sets, discover sum-product phenomena in Number Theory, and find a plethora of finite configurations in fractal sets. We will also extend the structure theory of ....Additive combinatorics of infinite sets via ergodic theoretic approach. The proposed project will utilise innovative ergodic theoretic approaches to enable us to address important questions in Additive Combinatorics (Number Theory) and Fractal Geometry. In particular, we will resolve long-standing inverse additive problems for infinite sets, discover sum-product phenomena in Number Theory, and find a plethora of finite configurations in fractal sets. We will also extend the structure theory of one of the most popular mathematical models of quasi-crystals to a more extensive class of groups. This project will make significant contributions to Additive Combinatorics and Ergodic Theory and will bring the Australian research in these fields to ever greater heights.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
Measure theoretic frameworks for limsup sets. This project aims to develop new powerful measure theoretic techniques in mathematics that will be used in establishing some indispensable results in analytical number theory (Diophantine approximation) and dynamical systems. The plan is to construct new techniques and to use them in situations where existing techniques are not applicable. As a consequence of the proposed frameworks, not only we aim to resolve a few long-standing problems such as the ....Measure theoretic frameworks for limsup sets. This project aims to develop new powerful measure theoretic techniques in mathematics that will be used in establishing some indispensable results in analytical number theory (Diophantine approximation) and dynamical systems. The plan is to construct new techniques and to use them in situations where existing techniques are not applicable. As a consequence of the proposed frameworks, not only we aim to resolve a few long-standing problems such as the Generalised Baker-Schmidt Problem (1970) but also envisage that the proposed frameworks will have far-reaching applications beyond the confines of Diophantine approximation and dynamical systems, for example, geometric measure theory, geometric probability and stochastic geometry etc. Read moreRead less