Discovery Early Career Researcher Award - Grant ID: DE200101834
Funder
Australian Research Council
Funding Amount
$418,410.00
Summary
The structure of singularities in geometric flows. The proposed research aims to develop our understanding of the structure of singularities in mean curvature and related flows, with certain applications in mind.
Discovery Early Career Researcher Award - Grant ID: DE210100535
Funder
Australian Research Council
Funding Amount
$340,548.00
Summary
Minimal surfaces and singularities of mean curvature flow. The project aims to characterise the geometric structure of minimal surfaces in the variational theory and classify singularities of mean curvature flow. Minimal surfaces are mathematical models of soap films, and their time-varying analogue is mean curvature flow, a dynamic process by which a surface flows to decrease its area as quickly as possible. As a central topic in geometric analysis, the theory of minimal surfaces and mean curv ....Minimal surfaces and singularities of mean curvature flow. The project aims to characterise the geometric structure of minimal surfaces in the variational theory and classify singularities of mean curvature flow. Minimal surfaces are mathematical models of soap films, and their time-varying analogue is mean curvature flow, a dynamic process by which a surface flows to decrease its area as quickly as possible. As a central topic in geometric analysis, the theory of minimal surfaces and mean curvature flow has proven to be a powerful and essential tool in mathematics. The project expects to generate new and significant results in minimal surfaces and singularity analysis of mean curvature flow and enhance potential applications in related disciplines such as computer vision and probability.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE180100110
Funder
Australian Research Council
Funding Amount
$343,450.00
Summary
Analysis of fully non-linear geometric problems and differential equations. This project aims to investigate non-linear geometric evolution equations that have received considerable attention in the past decades through their use in solving outstanding problems in mathematics, such as the Poincare conjecture. By developing innovative new techniques intertwining geometry and analysis, the project endeavours to make advances in non-linear problems modelling complex phenomena. The project addresses ....Analysis of fully non-linear geometric problems and differential equations. This project aims to investigate non-linear geometric evolution equations that have received considerable attention in the past decades through their use in solving outstanding problems in mathematics, such as the Poincare conjecture. By developing innovative new techniques intertwining geometry and analysis, the project endeavours to make advances in non-linear problems modelling complex phenomena. The project addresses topics as varied as hyperbolic geometry, and a geometric approach to irregularities forming in crystal growth in materials science, focusing on developing cutting-edge mathematical tools and connections to geometry.Read moreRead less
Parabolic methods for elliptic boundary value problems. This project aims to uncover new results for second order nonlinear elliptic partial differential equations via the use of uniqueness properties of solutions for related nonlinear parabolic partial differential equations. This will build on theory for fully nonlinear equations developed over the last 30 years. The project expects to generate new knowledge in the theory that will guide future research and have direct impact to applications ....Parabolic methods for elliptic boundary value problems. This project aims to uncover new results for second order nonlinear elliptic partial differential equations via the use of uniqueness properties of solutions for related nonlinear parabolic partial differential equations. This will build on theory for fully nonlinear equations developed over the last 30 years. The project expects to generate new knowledge in the theory that will guide future research and have direct impact to applications in optimal transport, geometric problems and more applied areas including image analysis and mathematical finance. The project will enhance Australia's international reputation for research in the field and train some of the next generation of mathematical analysts.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE230100415
Funder
Australian Research Council
Funding Amount
$422,154.00
Summary
Rigidity and boundary phenomena for geometric variational problems. The proposed project aims to investigate theoretical properties of thin films and fluid interfaces, which are modelled as surfaces driven by surface tension, possibly in an enclosing container. This project is expected to generate new knowledge in the area of geometric partial differential equations, by utilising new techniques in geometric flows, and by establishing novel methods for boundary value problems. The developed techn ....Rigidity and boundary phenomena for geometric variational problems. The proposed project aims to investigate theoretical properties of thin films and fluid interfaces, which are modelled as surfaces driven by surface tension, possibly in an enclosing container. This project is expected to generate new knowledge in the area of geometric partial differential equations, by utilising new techniques in geometric flows, and by establishing novel methods for boundary value problems. The developed techniques may have far-reaching applications in other areas of mathematical analysis, and the expected results would contribute greatly to the theory of surfaces governed by mean curvature, which arise in various real-world phenomena such as soap bubbles, black hole horizons and bushfire fronts. Read moreRead less
Australian Laureate Fellowships - Grant ID: FL150100126
Funder
Australian Research Council
Funding Amount
$2,080,100.00
Summary
Geometric analysis of eigenvalues and heat flows. Geometric analysis of eigenvalues and heat flows: This fellowship project aims to build on Australia's leading position in the areas of nonlinear partial differential equations and geometric analysis to exploit new and highly innovative mathematical methods. It is expected that the methods will affect a range of related fields including stochastic modelling and finance, image processing, and the basic sciences. The project seeks to serve as a foc ....Geometric analysis of eigenvalues and heat flows. Geometric analysis of eigenvalues and heat flows: This fellowship project aims to build on Australia's leading position in the areas of nonlinear partial differential equations and geometric analysis to exploit new and highly innovative mathematical methods. It is expected that the methods will affect a range of related fields including stochastic modelling and finance, image processing, and the basic sciences. The project seeks to serve as a focal point for a developing community of Australian researchers in this field, providing a training ground for young researchers and students at the forefront of a vigorous and internationally active area of research, and bringing top international researchers to Australia to interact with the local research community.Read moreRead less
Nuclear RNA surveillance and its connection to splicing quality control. Due to the error-prone nature of RNA splicing, elaborate quality control processes ensure that only correctly spliced transcripts can leave the nucleus. It has long been known that incorrectly spliced mRNA transcripts are degraded by the nuclear RNA surveillance machinery, but how the RNA quality control machinery is connected to nuclear RNA surveillance is not known. This proposal aims to uncover the connection between the ....Nuclear RNA surveillance and its connection to splicing quality control. Due to the error-prone nature of RNA splicing, elaborate quality control processes ensure that only correctly spliced transcripts can leave the nucleus. It has long been known that incorrectly spliced mRNA transcripts are degraded by the nuclear RNA surveillance machinery, but how the RNA quality control machinery is connected to nuclear RNA surveillance is not known. This proposal aims to uncover the connection between these two important processes and will fill a significant gap in our understanding of how splicing quality control and nuclear RNA surveillance work. The project will also identify sequence features that trigger abortive splicing reactions and will thus help to improve the design of synthetic mRNAs.Read moreRead less
Singularity and regularity for Monge-Ampere type equations. The Monge-Ampere equation, as a premier nonlinear partial differential equation, arises in several areas including geometry, physics, and optimal transportation. Many important problems and applications are related to the regularity of solutions, which are obstructed by singularities. This project aims to classify the geometry of the singular sets, and to establish a comprehensive regularity theory for general Monge-Ampere type equation ....Singularity and regularity for Monge-Ampere type equations. The Monge-Ampere equation, as a premier nonlinear partial differential equation, arises in several areas including geometry, physics, and optimal transportation. Many important problems and applications are related to the regularity of solutions, which are obstructed by singularities. This project aims to classify the geometry of the singular sets, and to establish a comprehensive regularity theory for general Monge-Ampere type equations by using innovative approaches and developing cutting-edge technologies in partial differential equations. Expected outcomes include the resolution of outstanding open problems. This project will significantly enhance Australia’s leadership and expertise in a major area of mathematics and applications.Read moreRead less
Role of R-loops and double R-loops in genome organisation and transcription. The majority of our genome is converted to an extensive network of non-protein-coding RNA molecules (ncRNAs), but the function of these ncRNAs is unknown. This project aims to identify and determine the mechanism of action of nuclear ncRNA networks with a particular focus on nuclear ncRNAs that form RNA-DNA hybrids with the genomic DNA. These studies have the potential to lead to ground-breaking discoveries in our under ....Role of R-loops and double R-loops in genome organisation and transcription. The majority of our genome is converted to an extensive network of non-protein-coding RNA molecules (ncRNAs), but the function of these ncRNAs is unknown. This project aims to identify and determine the mechanism of action of nuclear ncRNA networks with a particular focus on nuclear ncRNAs that form RNA-DNA hybrids with the genomic DNA. These studies have the potential to lead to ground-breaking discoveries in our understanding of genome organisation and the mechanism of transcription control, and might provide an entirely new tool-box to manipulate genome function. This should provide significant benefits to efforts to develop innovative biotechnology and genome editing technologies in plants and animals.Read moreRead less
RNA surveillance and the initial steps of RNA biogenesis. This project aims to understand the initial steps of RNA biogenesis and how this process is linked to the chromatin environment. Although less than five per cent of our genome encodes proteins, almost the entire genome is transcribed to RNA. A large portion of these transcripts are degraded during the early steps of RNA biogenesis by the RNA surveillance machinery, but the mechanism for the recognition and degradation of these transcripts ....RNA surveillance and the initial steps of RNA biogenesis. This project aims to understand the initial steps of RNA biogenesis and how this process is linked to the chromatin environment. Although less than five per cent of our genome encodes proteins, almost the entire genome is transcribed to RNA. A large portion of these transcripts are degraded during the early steps of RNA biogenesis by the RNA surveillance machinery, but the mechanism for the recognition and degradation of these transcripts is not understood. New evidence suggests that the chromatin environment of the transcribed locus plays an important role in this process. This project will lead to significant benefits in the implementation of emerging RNA-based technologies and in understanding how genome stability is maintained.Read moreRead less