Geometric evolution of spaces with symmetries. Symmetries underpin numerous laws of nature and mathematical constructions. This project aims to develop a comprehensive theory of the famous Ricci flow equation in the presence of symmetries. Previous study of this equation has led to many ground-breaking results, such as Perelman's celebrated proof of the century-old Poincaré conjecture. Outcomes are expected to fill major knowledge gaps in mathematics, opening doors to applications in quantum fie ....Geometric evolution of spaces with symmetries. Symmetries underpin numerous laws of nature and mathematical constructions. This project aims to develop a comprehensive theory of the famous Ricci flow equation in the presence of symmetries. Previous study of this equation has led to many ground-breaking results, such as Perelman's celebrated proof of the century-old Poincaré conjecture. Outcomes are expected to fill major knowledge gaps in mathematics, opening doors to applications in quantum field theory, relativity and other fields. Anticipated benefits include strengthening Australia’s leadership in mathematical innovation, advancing the internationalisation of the Australian research scene, and increasing the involvement of women in STEM.Read moreRead less
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: DE190101063
Funder
Australian Research Council
Funding Amount
$319,474.00
Summary
Geometric flows and distinguished geometric structures with symmetry. This project aims to carry out analysis for the Ricci flow and a number of recent flows in complex geometry. Geometric evolution equations are a trending topic in modern mathematics, with successful applications such as the celebrated proof of the century-old Poincaré Conjecture in 2002-03 by means of the Ricci flow. An analysis of the behaviour of solutions with symmetry is an essential component in any mature theory on the s ....Geometric flows and distinguished geometric structures with symmetry. This project aims to carry out analysis for the Ricci flow and a number of recent flows in complex geometry. Geometric evolution equations are a trending topic in modern mathematics, with successful applications such as the celebrated proof of the century-old Poincaré Conjecture in 2002-03 by means of the Ricci flow. An analysis of the behaviour of solutions with symmetry is an essential component in any mature theory on the subject. The project will use the novel tools and ideas resulting from this analysis to address the Alekseevskii Conjecture, a 40-year old fundamental open question in the field with major ramifications in mathematics and beyond. This project will further reinforce Australia's leading role in geometric analysis.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE220100919
Funder
Australian Research Council
Funding Amount
$345,000.00
Summary
Distinguished Geometric Structures with Symmetry in Four Dimensions. The Ricci flow is a geometric evolution equation having significant applications in geometry, topology, as well as in physics, biology and image processing. This project aims to provide a complete description and classification of highly symmetric, self-similar solutions to the Ricci Flow in four dimensions. Such a classification is essential to understanding the behaviour of the flow, but has so far evaded discovery. This proj ....Distinguished Geometric Structures with Symmetry in Four Dimensions. The Ricci flow is a geometric evolution equation having significant applications in geometry, topology, as well as in physics, biology and image processing. This project aims to provide a complete description and classification of highly symmetric, self-similar solutions to the Ricci Flow in four dimensions. Such a classification is essential to understanding the behaviour of the flow, but has so far evaded discovery. This project intends to combine techniques from pure mathematics with computational techniques to complete this classification. Such an outcome would greatly improve the understanding of the geometry of four-dimensional manifolds, potentially leading to applications in several areas of science as well as image processing.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.
New mathematics for understanding complex patterns in the natural sciences. This project aims to examine the interaction of fundamental two-dimensional patterns such as spots and stripes in reaction-diffusion equations, by developing and extending mathematical techniques. These fundamental planar structures form the backbone of more complex patterns and are, for example, observed in models that describe the propagation of impulses in nerve axons and the formation of vegetation patterns. The futu ....New mathematics for understanding complex patterns in the natural sciences. This project aims to examine the interaction of fundamental two-dimensional patterns such as spots and stripes in reaction-diffusion equations, by developing and extending mathematical techniques. These fundamental planar structures form the backbone of more complex patterns and are, for example, observed in models that describe the propagation of impulses in nerve axons and the formation of vegetation patterns. The future impact of this research will have economic and environmental benefits. For example, the project will develop a deeper understanding of interacting patterns that will provide insights into the role of vegetation in ecosystems that are undergoing desertification.Read moreRead less
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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