The Ricci curvature of homogeneous spaces. The geometry of homogeneous spaces is an area of research with applications in numerous fields, including topology, harmonic analysis, relativity and quantum theory. This project aims to resolve a fundamental problem in this area, known as the prescribed Ricci curvature problem for homogeneous metrics, and to settle the important and closely related question of Ricci iteration existence and convergence. Moreover, the project aims to exploit the interpla ....The Ricci curvature of homogeneous spaces. The geometry of homogeneous spaces is an area of research with applications in numerous fields, including topology, harmonic analysis, relativity and quantum theory. This project aims to resolve a fundamental problem in this area, known as the prescribed Ricci curvature problem for homogeneous metrics, and to settle the important and closely related question of Ricci iteration existence and convergence. Moreover, the project aims to exploit the interplay between geometry and algebra to provide new insight into the physically significant problem of classifying unitary Lie algebra representations. This project is expected to facilitate interdisciplinary interaction leading to exciting developments across a range of fields.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE150101548
Funder
Australian Research Council
Funding Amount
$345,000.00
Summary
Geometric boundary-value problems. The Ricci flow is a geometric differential equation which recently made headlines for its key role in the proof of the Poincaré Conjecture (a century-old mathematical conjecture whose resolution carried a $1,000,000 prize). Developing the theory of boundary-value problems for the Ricci flow is a fundamental question which has remained open for over two decades. This project aims to answer this question on a wide class of spaces, along with the closely related q ....Geometric boundary-value problems. The Ricci flow is a geometric differential equation which recently made headlines for its key role in the proof of the Poincaré Conjecture (a century-old mathematical conjecture whose resolution carried a $1,000,000 prize). Developing the theory of boundary-value problems for the Ricci flow is a fundamental question which has remained open for over two decades. This project aims to answer this question on a wide class of spaces, along with the closely related question of solvability of boundary-value problems for the prescribed Ricci curvature equation. The results will have ramifications in a variety of fields, from pure mathematics to quantum field theory, relativity and modelling of biological systems.Read moreRead less
ARC Complex Open Systems Research Network. Complexity is the common frontier in the physical, biological and social sciences. This Network will link specialists in all three sciences through five generic conceptual and mathematical theme activities. It will promote research into how subsystems self-organise into new emergent structures when assembled into an open, non-equilibrium system. Outcomes will include new technologies and software tools and deeper understanding of fundamental questions i ....ARC Complex Open Systems Research Network. Complexity is the common frontier in the physical, biological and social sciences. This Network will link specialists in all three sciences through five generic conceptual and mathematical theme activities. It will promote research into how subsystems self-organise into new emergent structures when assembled into an open, non-equilibrium system. Outcomes will include new technologies and software tools and deeper understanding of fundamental questions in science. An essential function of the network will be introducing researchers end users to new tools and broadening the horizons of graduate students.Read moreRead less
Dissipation and relaxation in statistical mechanics. This project studies the mathematical conditions for relaxation either to equilibrium or to steady states, which is important in predicting behaviour in diverse fields including climate modelling, materials science, nanotechnology and biology. Early career researchers will be involved in the project, gaining valuable skills in theory and simulation.
Special Research Initiatives - Grant ID: SR0354741
Funder
Australian Research Council
Funding Amount
$10,000.00
Summary
Quantum Many-Body Systems Network: Breakthrough Science and Frontier Technologies. This Initiative will bring together leading researchers with complementary expertise in mathematics and the enabling sciences to form a Network fostering world leading fundamental research and innovation in quantum many-body systems. The collaborative effort between mathematicians with powerful and sophisticated new techniques and physicists and chemists with deep insight into the challenges and opportunities of t ....Quantum Many-Body Systems Network: Breakthrough Science and Frontier Technologies. This Initiative will bring together leading researchers with complementary expertise in mathematics and the enabling sciences to form a Network fostering world leading fundamental research and innovation in quantum many-body systems. The collaborative effort between mathematicians with powerful and sophisticated new techniques and physicists and chemists with deep insight into the challenges and opportunities of the quantum realm will lead to breakthrough science of vital importance to the development of frontier technologies in Australia. This Network will also place a strong emphasis on research training, the mentoring of early career researchers and establishing collaborations with leading international research groups and networks.
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Properties of nonequilibrium steady states. A nonequilibrium steady state (NESS) occurs when work is performed on a system and the heat so generated is absorbed by a thermostatting mechanism. The system settles into steady state and its properties no longer change. Almost all experimental systems of interest are in a nonequilibrium state, so understanding NESSs is highly significant. Unlike time stationary equilibrium states, the distribution of microstates in a NESS cannot be described by simpl ....Properties of nonequilibrium steady states. A nonequilibrium steady state (NESS) occurs when work is performed on a system and the heat so generated is absorbed by a thermostatting mechanism. The system settles into steady state and its properties no longer change. Almost all experimental systems of interest are in a nonequilibrium state, so understanding NESSs is highly significant. Unlike time stationary equilibrium states, the distribution of microstates in a NESS cannot be described by simple closed form distributions. This project will determine properties, symmetries and extrema of NESS using concepts and theorems developed for studying transient nonequilibrium states, and will also determine if approximate, physically relevant forms of the phase space distributions can be developed.Read moreRead less
Algebraic Structures in Mathematical Physics and Their Applications. Algebraic structures such as affine (super)algebras, quantised algebras and vertex operator algebras are among the most important discoveries in mathematics. They provide a universal common algebraic framework underlying applications in a wide range of physics (eg. statistical mechanics, string theory, condensed matter physics etc.) leading to a high level of research activity worldwide. The project harnessess the high level ....Algebraic Structures in Mathematical Physics and Their Applications. Algebraic structures such as affine (super)algebras, quantised algebras and vertex operator algebras are among the most important discoveries in mathematics. They provide a universal common algebraic framework underlying applications in a wide range of physics (eg. statistical mechanics, string theory, condensed matter physics etc.) leading to a high level of research activity worldwide. The project harnessess the high level of expertise in mathematical physics across Australia to focus on exciting new developments in the theory of these algebraic structures and their application to physics, thus ensuring Australia plays a leading role in this rapidly expanding field.Read moreRead less
Reaching new frontiers of quantum fields and gravity through deformations. This project aims to reach new frontiers in quantum field and gravity theories. These underpin systems ranging from semi-conductors to particle collisions and the quantum behavior of black holes. An obstacle is that these theories are notoriously hard to solve. This project proposes to tackle this longstanding problem by using new deformations, symmetries and dualities that have attracted widespread attention. Expected ou ....Reaching new frontiers of quantum fields and gravity through deformations. This project aims to reach new frontiers in quantum field and gravity theories. These underpin systems ranging from semi-conductors to particle collisions and the quantum behavior of black holes. An obstacle is that these theories are notoriously hard to solve. This project proposes to tackle this longstanding problem by using new deformations, symmetries and dualities that have attracted widespread attention. Expected outcomes will include innovative techniques that will greatly enhance and interconnect our knowledge of field theories and quantum gravity, together with new discoveries in quantum-corrected geometries. A new network of domestic and international experts will largely benefit the fields of theoretical and mathematical physics.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
Centre for Mathematical and Statistical Modelling of Complex Systems. This Centre, formed by a group of high-profile researchers, brings expertise from linked but hitherto disparate areas together. It will place Australia at the forefront of research into complex systems.
The mission of the Centre is to stimulate research in mathematical and statistical modelling of complex systems and to encourage cross-fertilisation of ideas and techniques. The specific objectives are
- to formulate and ana ....Centre for Mathematical and Statistical Modelling of Complex Systems. This Centre, formed by a group of high-profile researchers, brings expertise from linked but hitherto disparate areas together. It will place Australia at the forefront of research into complex systems.
The mission of the Centre is to stimulate research in mathematical and statistical modelling of complex systems and to encourage cross-fertilisation of ideas and techniques. The specific objectives are
- to formulate and analyse mathematical and statistical models for natural and artificial complex systems,
- to use these models to develop an understanding of the behaviour of these systems
- to incorporate this understanding into strategies for management and control.Read moreRead less