Geometric evolution problems in nonlinear partial differential equations. This project aims to address important problems key to the understanding of geometric evolution equations and certain other nonlinear partial differential equations. The problems to be tackled lie in a very active area of mathematics: harmonic maps, liquid crystals and Yang-Mills theory. Special aims are to exploit new methods to settle open problems in harmonic maps and Yang-Mills equations, and to improve understanding o ....Geometric evolution problems in nonlinear partial differential equations. This project aims to address important problems key to the understanding of geometric evolution equations and certain other nonlinear partial differential equations. The problems to be tackled lie in a very active area of mathematics: harmonic maps, liquid crystals and Yang-Mills theory. Special aims are to exploit new methods to settle open problems in harmonic maps and Yang-Mills equations, and to improve understanding of practical questions such as the mathematical modelling of liquid crystals via the celebrated Ericksen-Leslie and Landau-de Gennes theories. The expected outcomes are fundamental results in mathematics, with applications in other sciences.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
The mathematics of novel magnetic memory materials. Magnetic memories are the world’s principal device for storing information. The next generation will have greatly increased access speed and data-storage capacity. This project will develop the mathematical theory of these new magnetic memory materials, a crucial first step in understanding and being able to fine-tune their properties.
Variational methods in partial differential equations. Research in partial differential equations is a very active area of modern mathematics linking nonlinear functional analysis, calculus of variations and differential geometry to applied sciences. This project will enable Australia-based researchers to participate in the forefront of mathematical research with leading international mathematicians by establishing new collaborations, strengthening on-going collaborations and providing internat ....Variational methods in partial differential equations. Research in partial differential equations is a very active area of modern mathematics linking nonlinear functional analysis, calculus of variations and differential geometry to applied sciences. This project will enable Australia-based researchers to participate in the forefront of mathematical research with leading international mathematicians by establishing new collaborations, strengthening on-going collaborations and providing international research experience for early career researchers. As a result, this proposal will enhance Australia's distinguished reputation in analysis and further link the UQ group with a number of mathematical institutes in USA and China.Read moreRead less
Geometric partial differential systems and their applications. This proposal addresses questions central to the understanding of nonlinear partial differential systems from classical, quantum field theory and liquid crystals. Applications to physical problems such as the Yang-Mills flow, Faddeev's model and liquid crystal systems are of great interest and importance in the broader scientific community. The project will yield internationally significant results in theoretical mathematics, with ....Geometric partial differential systems and their applications. This proposal addresses questions central to the understanding of nonlinear partial differential systems from classical, quantum field theory and liquid crystals. Applications to physical problems such as the Yang-Mills flow, Faddeev's model and liquid crystal systems are of great interest and importance in the broader scientific community. The project will yield internationally significant results in theoretical mathematics, with applications in physics and and other sciences. Specialist training will be provided for Australia's next generation of mathematicians. This project will enable Australian researchers to stay at the forefront of research in this area, strengthening links with a number of world-leading mathematicians.Read moreRead less
Geometric variational problems and nonlinear partial differential systems. We will investigate several important problems on non-linear partial differential systems, bridging analysis, differential geometry and mathematical physics. Harmonic maps are the prototype of maps minimizing the Dirichlet energy. The liquid crystal configuration generalizes the harmonic map with values into two dimensional spheres. The Yang-Mills equations originated from particle physics. We will make fundamental contri ....Geometric variational problems and nonlinear partial differential systems. We will investigate several important problems on non-linear partial differential systems, bridging analysis, differential geometry and mathematical physics. Harmonic maps are the prototype of maps minimizing the Dirichlet energy. The liquid crystal configuration generalizes the harmonic map with values into two dimensional spheres. The Yang-Mills equations originated from particle physics. We will make fundamental contributions to these topics: Regularity problem and energy minimality of weakly harmonic maps, Weak solutions of the liquid crystal equilibrium system, Yang-Mills heat flow and singular Yang-Mills connections.
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Discovery Early Career Researcher Award - Grant ID: DE140100741
Funder
Australian Research Council
Funding Amount
$389,564.00
Summary
Analysis of defect driven pattern formation in mathematical models. . Defects, or heterogeneities, are common in nature and technology and therefore in mathematical models. This project will underpin the effects a defect can have on the dynamics of a model, characterise the new patterns created by a heterogeneity and see how the dynamics can be controlled by manipulating the heterogeneity. Moreover, these new insights will be applied to a model for skin cancer, resulting in a more appropriate mo ....Analysis of defect driven pattern formation in mathematical models. . Defects, or heterogeneities, are common in nature and technology and therefore in mathematical models. This project will underpin the effects a defect can have on the dynamics of a model, characterise the new patterns created by a heterogeneity and see how the dynamics can be controlled by manipulating the heterogeneity. Moreover, these new insights will be applied to a model for skin cancer, resulting in a more appropriate model and a mathematically justifiable analysis of a very important scientific problem.Read moreRead less
Determination of Conductivity Values for Anisotropic Tissue. Well established mathematical models governing the electrical potential in biological tissue can be combined with measurements of the electric potential on the surface of the tissue to provide insight into subsurface tissue damage. However, before such observations can be convincingly accepted, reliable values for the tissue conductivity must be obtained. The aim of this project is to develop mathematical techniques to calculate the co ....Determination of Conductivity Values for Anisotropic Tissue. Well established mathematical models governing the electrical potential in biological tissue can be combined with measurements of the electric potential on the surface of the tissue to provide insight into subsurface tissue damage. However, before such observations can be convincingly accepted, reliable values for the tissue conductivity must be obtained. The aim of this project is to develop mathematical techniques to calculate the conductivity values so that one can apply the equations to solve problems of potential distribution and proceed to accurately simulate electrical potential distributions in damaged tissue. More accurate and reliable conductivity values will allow a better understanding of the way electric current moves through the heart which, in turn, will result in more efficient defibrillators and better diagnosis of abnormal function.Read moreRead less
Multi-scale modelling of cell migration in developmental biology. Interpretative and predictive tools are needed for the comprehensive understanding of directed cell migration in the medical sciences. Mathematical models and modelling methodologies developed in this project will make a significant contribution to the investigation of cell migration and the testing and generation of hypotheses. Such models are needed to understand observed cellular patterns. This project will contribute to knowle ....Multi-scale modelling of cell migration in developmental biology. Interpretative and predictive tools are needed for the comprehensive understanding of directed cell migration in the medical sciences. Mathematical models and modelling methodologies developed in this project will make a significant contribution to the investigation of cell migration and the testing and generation of hypotheses. Such models are needed to understand observed cellular patterns. This project will contribute to knowledge of normal and abnormal developmental processes, especially in embryonic growth. Understanding these processes should lead to prediction and treatment of congenital disorders and contribute to a healthy start to life.Read moreRead less
Statistical and mathematical modelling to improve health care outcomes in hospitals. The aim of this project is to develop new quantitative techniques based on mathematical and statistical modelling that improve the outcomes of health care in hospitals. Hospital outcomes for patients are sub-optimal due to adverse events such as hospital acquired infections and fully stretched facilities. Research from this project will lead to resource usage being optimised using operations research; the tra ....Statistical and mathematical modelling to improve health care outcomes in hospitals. The aim of this project is to develop new quantitative techniques based on mathematical and statistical modelling that improve the outcomes of health care in hospitals. Hospital outcomes for patients are sub-optimal due to adverse events such as hospital acquired infections and fully stretched facilities. Research from this project will lead to resource usage being optimised using operations research; the transmission of hospital acquired infections being better understood using mathematical models; and better monitoring of adverse events and analyses of studies using statistical tools. Opportunities will be provided for hospital staff to acquire knowledge of the significance of these outcomes .Read moreRead less