Dynamics on space-filling shapes. Modern science derives its power from mathematical models and tools that enable us to predict their behaviours. The project aims to construct new models given by dynamical systems that move consistently from one tile to another in a lattice of higher-dimensional shapes called polytopes. The construction is expected to lead to new functions with properties that will provide extensions of current models of growth processes. The intended outcomes of the project inc ....Dynamics on space-filling shapes. Modern science derives its power from mathematical models and tools that enable us to predict their behaviours. The project aims to construct new models given by dynamical systems that move consistently from one tile to another in a lattice of higher-dimensional shapes called polytopes. The construction is expected to lead to new functions with properties that will provide extensions of current models of growth processes. The intended outcomes of the project include predictive tools that describe nonlinear special functions and information about their symmetry reductions. This should provide significant benefits, such as new mathematical knowledge, innovative techniques, and enhanced scientific capacity in Australia.Read moreRead less
Non-local equations at work. This project aims to study non-local fractional equations. These problems arise naturally in many fields of pure and applied mathematics. This project will consider symmetry and rigidity results; problems from atom dislocation theory; nonlocal minimal surfaces; symbolic dynamics for nonlocal equations; and free boundary problems. This project aims to obtain substantial progress in this field, both from the point of view of the mathematical theory and in view of concr ....Non-local equations at work. This project aims to study non-local fractional equations. These problems arise naturally in many fields of pure and applied mathematics. This project will consider symmetry and rigidity results; problems from atom dislocation theory; nonlocal minimal surfaces; symbolic dynamics for nonlocal equations; and free boundary problems. This project aims to obtain substantial progress in this field, both from the point of view of the mathematical theory and in view of concrete applications. This project should contribute to the development of the mathematical theory and give insight for concrete applications in physics and biology.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
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
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: FL190100081
Funder
Australian Research Council
Funding Amount
$3,532,919.00
Summary
Minimal surfaces, free boundaries and partial differential equations. This project enhances Australia as a world leader in the field of mathematical analysis, focusing on regularity and qualitative properties of solutions of partial differential equations and nonlocal problems, and solving very challenging research questions in a key strategic area of international science.
The broad applicability of the results constitutes a very fertile ground for cross-disciplinary interactions with scientist ....Minimal surfaces, free boundaries and partial differential equations. This project enhances Australia as a world leader in the field of mathematical analysis, focusing on regularity and qualitative properties of solutions of partial differential equations and nonlocal problems, and solving very challenging research questions in a key strategic area of international science.
The broad applicability of the results constitutes a very fertile ground for cross-disciplinary interactions with scientists of other disciplines.
A new research team based in Western Australia will be founded, connecting world leaders and talented early career researchers, providing an ideal training environment for students and PostDocs, offering an excellent image of the scientific community and developing strategic fields of knowledge.Read moreRead less
Monge-Ampere equations and applications. The Monge-Ampere equation is a premier fully nonlinear partial differential equation with significant applications in geometry, physics and applied science. Building upon breakthroughs made by the proposers in previous grant research, this project aims to resolve challenging problems involving Monge-Ampere type equations and applications. The project goal is to establish new regularity theory and classify singularity profile for solutions to Monge-Ampere ....Monge-Ampere equations and applications. The Monge-Ampere equation is a premier fully nonlinear partial differential equation with significant applications in geometry, physics and applied science. Building upon breakthroughs made by the proposers in previous grant research, this project aims to resolve challenging problems involving Monge-Ampere type equations and applications. The project goal is to establish new regularity theory and classify singularity profile for solutions to Monge-Ampere type equation arising in applied sciences, by introducing new ideas and developing innovative cutting-edge techniques. Expected outcomes include resolution of outstanding open problems and continuing enhancement of Australian leadership and expertise in a major area of mathematics.
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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
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
Discovery Early Career Researcher Award - Grant ID: DE190101231
Funder
Australian Research Council
Funding Amount
$390,000.00
Summary
Integrable systems from gauge theories. This project aims to construct and describe three important integrable systems and to produce new significant results in representation theory and mathematical physics. The focus of the project is the theory of integrable systems, a breakthrough area in modern mathematics, and it will use tools from geometric representation theory to study the interrelation between geometry and mathematical physics. Originating from string theory, the project will not only ....Integrable systems from gauge theories. This project aims to construct and describe three important integrable systems and to produce new significant results in representation theory and mathematical physics. The focus of the project is the theory of integrable systems, a breakthrough area in modern mathematics, and it will use tools from geometric representation theory to study the interrelation between geometry and mathematical physics. Originating from string theory, the project will not only provide a deeper understanding of the universe, it will also train young mathematicians and other highly qualified individuals with the potential to make a significant impact to technology, security, and the economy though their specialised skills.Read moreRead less