Geodetic groups: foundational problems in algebra and computer science. The project aims to resolve important and longstanding open problems in Geometric Group Theory and Theoretical Computer Science. Since the 1980s researchers have conjectured that the geometric property of being geodetic is equivalent to several purely algebraic, algorithmic, and language-theoretic characterisations.
The project team's expertise in geodesic properties of groups, the interaction between formal languages and g ....Geodetic groups: foundational problems in algebra and computer science. The project aims to resolve important and longstanding open problems in Geometric Group Theory and Theoretical Computer Science. Since the 1980s researchers have conjectured that the geometric property of being geodetic is equivalent to several purely algebraic, algorithmic, and language-theoretic characterisations.
The project team's expertise in geodesic properties of groups, the interaction between formal languages and groups, and the theory of rewriting systems, together with recent breakthroughs by the team ensures that significant results can be expected.
Benefits include training research students and postdoctoral researchers in cutting-edge techniques, and advancing fundamental knowledge in mathematics and computer science.Read moreRead less
Tractable topological computing: Escaping the hardness trap. Computational topology is a young and energetic field that uses computers to solve complex geometric problems driven by pure mathematics, and with diverse applications in biology, signal processing and data mining. A major barrier is that many of these problems are thought to be fundamentally and intractably hard. This project aims to defy such barriers for typical real-world inputs by fusing geometric techniques with technologies from ....Tractable topological computing: Escaping the hardness trap. Computational topology is a young and energetic field that uses computers to solve complex geometric problems driven by pure mathematics, and with diverse applications in biology, signal processing and data mining. A major barrier is that many of these problems are thought to be fundamentally and intractably hard. This project aims to defy such barriers for typical real-world inputs by fusing geometric techniques with technologies from the field of parameterised complexity, creating powerful, practical solutions for these problems. It is expected to shed much-needed light on the vast and puzzling gap between theory and practice, and give researchers fast new software tools for large-scale experimentation and cutting-edge computer proofs.Read moreRead less
Unpacking the immune system with applied mathematics. This project aims to model immune interactions across cells and structures spanning scales of nanometres to millimetres. It expects to develop innovative mathematical insights, improve our understanding of immunology, and consolidate collaborations with top American and European laboratories and groups. Expected outcomes include cutting-edge techniques for multiscale biological modelling and improved prediction and analysis of immune dynami ....Unpacking the immune system with applied mathematics. This project aims to model immune interactions across cells and structures spanning scales of nanometres to millimetres. It expects to develop innovative mathematical insights, improve our understanding of immunology, and consolidate collaborations with top American and European laboratories and groups. Expected outcomes include cutting-edge techniques for multiscale biological modelling and improved prediction and analysis of immune dynamics. The project should provide benefits to industries where highly organised behaviours are important, for example those interested in robot swarming, optimal transportation, and epidemic management. It should also benefit Australian students and researchers with novel overseas training opportunities.Read moreRead less
A Novel Geometric Approach to Shocks in Reaction-Nonlinear Diffusion Models. Reaction-nonlinear diffusion models play a vital role in the study of cell migration and population dynamics. However, the presence of aggregation, or backward diffusion, leads to the formation of shock waves - distinct, sharp interfaces between different populations of densities of cells - and the breakdown of the model. This project will develop new geometric methods to explain the formation and temporal evolution of ....A Novel Geometric Approach to Shocks in Reaction-Nonlinear Diffusion Models. Reaction-nonlinear diffusion models play a vital role in the study of cell migration and population dynamics. However, the presence of aggregation, or backward diffusion, leads to the formation of shock waves - distinct, sharp interfaces between different populations of densities of cells - and the breakdown of the model. This project will develop new geometric methods to explain the formation and temporal evolution of these shock waves, while simultaneously unifying existing regularisation techniques under a single, geometric banner. It will devise innovative tools in singular perturbation theory and stability analysis that will identify key parameters in the creation of shock waves, as well as their dynamic behaviour.Read moreRead less
ARC Centre of Excellence for the Mathematical Analysis of Cellular Systems. ARC Centre of Excellence for the Mathematical Analysis of Cellular Systems. The ARC Centre for the Mathematical Analysis of Cellular Systems aims to deliver the mathematics required to compute life. The Centre will deliver innovation in computational and mathematical biology and establish in silico biology alongside in vivo and in vitro biology. These models will allow us to understand the complexity of life at the cellu ....ARC Centre of Excellence for the Mathematical Analysis of Cellular Systems. ARC Centre of Excellence for the Mathematical Analysis of Cellular Systems. The ARC Centre for the Mathematical Analysis of Cellular Systems aims to deliver the mathematics required to compute life. The Centre will deliver innovation in computational and mathematical biology and establish in silico biology alongside in vivo and in vitro biology. These models will allow us to understand the complexity of life at the cellular level and enable new ways of combining diverse and heterogenous data. This will allow us to understand the mechanisms underlying cellular behaviour, and to apply rational design engineering methods in order to control the dynamics of biological systems. Read moreRead less
Enabling students' critical mathematical thinking. The capacity to use mathematics critically is essential for making prudent decisions and forming balanced judgements about economic, health, environmental and other challenges facing society. Developing Critical Mathematical Thinking (CMT) in the classroom provides students with the necessary skills to address complex real-world problems. Fostering CMT, however, is difficult and teaching practices around its development are under-researched and ....Enabling students' critical mathematical thinking. The capacity to use mathematics critically is essential for making prudent decisions and forming balanced judgements about economic, health, environmental and other challenges facing society. Developing Critical Mathematical Thinking (CMT) in the classroom provides students with the necessary skills to address complex real-world problems. Fostering CMT, however, is difficult and teaching practices around its development are under-researched and under-theorised. This study aims to generate new insight into teaching practices that can promote or inhibit students' CMT development. To address this aim, we use an innovative video-based methodology that integrates researcher and teacher perspectives on students' CMT development.Read moreRead less
Reasonable Adjustments to Maths for Students with Intellectual Disabilities. Reasonable adjustments to secondary mathematics – parents want it, students need it, the Australian law requires it – but how do teachers achieve it? The aim of this project is to find ways to adjust, teach and assess secondary mathematics for students with intellectual and developmental disabilities. The significance of this study is in solving a problem facing teachers who are required by law to teach mathematics with ....Reasonable Adjustments to Maths for Students with Intellectual Disabilities. Reasonable adjustments to secondary mathematics – parents want it, students need it, the Australian law requires it – but how do teachers achieve it? The aim of this project is to find ways to adjust, teach and assess secondary mathematics for students with intellectual and developmental disabilities. The significance of this study is in solving a problem facing teachers who are required by law to teach mathematics without evidence informed approaches. Expected outcomes are an evidence-base for reasonable adjustments to secondary mathematics and approaches and strategies for teachers for inclusive practices. If ways for teaching year-level mathematics to learners with intellectual disability are developed, many more students may benefit.
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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
Developing feasible in situ control of mange disease in wombats. Our goal is the development of feasible in situ control of sarcoptic mange in wombat populations. Globally important, the Sarcoptes scabiei mite infects >100 mammal species and is among the 50 most common human diseases, causing health, welfare and population impacts. This infection is treatable, and we will test a new treatment (fluralaner), develop new models to guide management, and conduct replicated field trials. This will ena ....Developing feasible in situ control of mange disease in wombats. Our goal is the development of feasible in situ control of sarcoptic mange in wombat populations. Globally important, the Sarcoptes scabiei mite infects >100 mammal species and is among the 50 most common human diseases, causing health, welfare and population impacts. This infection is treatable, and we will test a new treatment (fluralaner), develop new models to guide management, and conduct replicated field trials. This will enable science-based guidelines, advancing disease control, local eradication, and regulatory approval for wombats. Our research framework is adaptable to other mange-impacted species, and advance methods and theory for control of treatable disease in wildlife.Read moreRead less
Mathematical and Numerical Models of Piezoelectric Wave Energy Converters. The project will investigate piezoelectric wave energy converters. We will derive the equations of motion in a form suitable for use in marine engineering paradigms using variational methods and then solve these analytically and with smoothed particle hydrodynamics. Using these innovative techniques, this project will generate new knowledge capable of elucidating the multifaceted physical phenomena that occur when comple .... Mathematical and Numerical Models of Piezoelectric Wave Energy Converters. The project will investigate piezoelectric wave energy converters. We will derive the equations of motion in a form suitable for use in marine engineering paradigms using variational methods and then solve these analytically and with smoothed particle hydrodynamics. Using these innovative techniques, this project will generate new knowledge capable of elucidating the multifaceted physical phenomena that occur when complex fluid motion and deformable structures interact. The project outcomes include the development of mathematical and computation methods to handle intricate behaviours of piezoelectric elastic-fluids systems. These groundbreaking methods will allow these wave energy systems to be analysed and their effectiveness assessed.Read moreRead less