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
Discovery Early Career Researcher Award - Grant ID: DE180100957
Funder
Australian Research Council
Funding Amount
$339,328.00
Summary
Partial differential equations, free boundaries and applications. This project aims to investigate fundamental problems in the analysis of partial differential equations and free boundary theory, to develop advanced mathematical theories with the possibility of important applications. The expected outcome is the establishment of a regularity and classification theory for nonlocal equations and for free boundary problems in linear and nonlinear settings. The benefit of the project lies in a concr ....Partial differential equations, free boundaries and applications. This project aims to investigate fundamental problems in the analysis of partial differential equations and free boundary theory, to develop advanced mathematical theories with the possibility of important applications. The expected outcome is the establishment of a regularity and classification theory for nonlocal equations and for free boundary problems in linear and nonlinear settings. The benefit of the project lies in a concrete advancement of the mathematical research with advantages for a deeper understanding of complex phenomena in physics and biology. Some of the problems also provide results useful for industrial applications.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
Group actions: combinatorics, geometry and computation. Science today relies on digital technologies using quantised and digital information. Because of the discrete nature of digital information, much of the mathematics underpinning these advances comes from the core disciplines of algebra and combinatorics within which this proposal falls. All aspects of the proposal focus on strengthening theoretical understanding of algebraic and combinatorial structures, and increasing computational power f ....Group actions: combinatorics, geometry and computation. Science today relies on digital technologies using quantised and digital information. Because of the discrete nature of digital information, much of the mathematics underpinning these advances comes from the core disciplines of algebra and combinatorics within which this proposal falls. All aspects of the proposal focus on strengthening theoretical understanding of algebraic and combinatorial structures, and increasing computational power for working with them. The fundamental research outcomes, in terms of theorems, algorithms, and the training of young research mathematicians, will thus both enhance the high international standing of Australian mathematics, and strengthen Australia's capabilities in these important areas.Read moreRead less
Groups: statistics, structure, and algorithms. Science today relies on digital technologies using quantised and digital information. Because of the discrete nature of digital information, much of the mathematics underpinning these advances comes from the core disciplines of algebra and combinatorics within which this proposal falls. All aspects of the proposal focus on strengthening theoretical understanding of algebraic and combinatorial structures, and increasing computational power for workin ....Groups: statistics, structure, and algorithms. Science today relies on digital technologies using quantised and digital information. Because of the discrete nature of digital information, much of the mathematics underpinning these advances comes from the core disciplines of algebra and combinatorics within which this proposal falls. All aspects of the proposal focus on strengthening theoretical understanding of algebraic and combinatorial structures, and increasing computational power for working with them. The fundamental research outcomes, in terms of theorems, algorithms, and the training of young research mathematicians, will thus both enhance the high international standing of Australian mathematics, and strengthen Australia's capabilities in these important areas.Read moreRead less
Factorisation of Finite Groups and Graphs. The combinatorial structure of a graph is strongly influenced by its
symmetry, and the symmetry is described precisely by its group of
automorphisms. Interplay between actions of the automorphism group on
vertices, edges, and other configurations, reveals important graph
structure, especially the existence of graph factorisations. In turn, a group factorisation arises whenever a group has two
independent transitive actions, and these arise in parti ....Factorisation of Finite Groups and Graphs. The combinatorial structure of a graph is strongly influenced by its
symmetry, and the symmetry is described precisely by its group of
automorphisms. Interplay between actions of the automorphism group on
vertices, edges, and other configurations, reveals important graph
structure, especially the existence of graph factorisations. In turn, a group factorisation arises whenever a group has two
independent transitive actions, and these arise in particular while
determining graph automorphism groups, and graph factorisations. We will classify families of group factorisations, especially for simple groups, and apply this to establish a theory of symmetrical graph
factorisations, and to study Cayley graphs and 2-closures of permutation groups.
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Efficient computation in finite groups with applications in algebra and graph theory. The cutting-edge research of the project will further strengthen Australia's prominent role in computational group theory and algebraic graph theory. Besides the theoretical advances, the project includes the implementation and wide distribution of matrix group algorithms, benefiting immediately the algebraic research community and undergraduate mathematical education.
Finite permutation groups and flag-transitive incidence structures. Mathematics is the enabling discipline for all the sciences and so a strong mathematical research community in Australia provides the foundations for future discoveries in science and technology. By developing new theory for permutation groups, producing a new paradigm for the study of Buekenhout geometries and classifying certain families of flag-transitive incidence structures, we will enhance Australia's leading position in P ....Finite permutation groups and flag-transitive incidence structures. Mathematics is the enabling discipline for all the sciences and so a strong mathematical research community in Australia provides the foundations for future discoveries in science and technology. By developing new theory for permutation groups, producing a new paradigm for the study of Buekenhout geometries and classifying certain families of flag-transitive incidence structures, we will enhance Australia's leading position in Permutation Group Theory, Algebraic Graph Theory and Finite Geometry. This will attract international and Australian postgraduate students and visitors, and strengthen the research activities of Australia by enhancing the collaboration between UWA and leading international universities.Read moreRead less
Group algorithms: Complexity, Theory and Practice. The symmetry of a mathematical or physical system is often best described by an abstract structure called a group, and groups are commonly represented as groups of permutations or matrices. In this project we shall design and analyse a general algorithmic framework for computing with finite groups. In the context of permutation groups and matrix groups we will produce prototype implementations. The proposed framework has the potential to revolut ....Group algorithms: Complexity, Theory and Practice. The symmetry of a mathematical or physical system is often best described by an abstract structure called a group, and groups are commonly represented as groups of permutations or matrices. In this project we shall design and analyse a general algorithmic framework for computing with finite groups. In the context of permutation groups and matrix groups we will produce prototype implementations. The proposed framework has the potential to revolutionise algorithmic group theory as it draws together theoretical and computational models of groups.Read moreRead less
Computing with large groups: probability distributions and fast randomised algorithms. Fast algorithms produced by the project will impact on the practical management of symmetry in large scale searches, which have important industrial applications. Hence the project addresses the Priority Goals Breakthrough Science and Smart Information Use. The project will enhance Australia's leading position in Computational Algebra. Implementations of our algorithms will be incorporated in the Computer Alge ....Computing with large groups: probability distributions and fast randomised algorithms. Fast algorithms produced by the project will impact on the practical management of symmetry in large scale searches, which have important industrial applications. Hence the project addresses the Priority Goals Breakthrough Science and Smart Information Use. The project will enhance Australia's leading position in Computational Algebra. Implementations of our algorithms will be incorporated in the Computer Algebra system Magma, based at the University of Sydney, distributed world-wide, and used intensively in research and teaching. The project will attract international and Australian graduate students and postdoctoral researchers, and strengthen research activities in Australia by enhancing already strong international collaborations. Read moreRead less