Symmetry in Differential Geometry. Differential geometry is a major branch of mathematics studying shape by using calculus and differential equations. This is a fundamental research project in this area, especially concerned with the interaction between geometry, differential equations, and symmetry. The mathematical notion of symmetry was already formalised early last century and nowadays lies at the very heart of mathematics and physics. Advances in this area provide essential tools in basic s ....Symmetry in Differential Geometry. Differential geometry is a major branch of mathematics studying shape by using calculus and differential equations. This is a fundamental research project in this area, especially concerned with the interaction between geometry, differential equations, and symmetry. The mathematical notion of symmetry was already formalised early last century and nowadays lies at the very heart of mathematics and physics. Advances in this area provide essential tools in basic science and unexpected technological benefits can easily arise (for example, in medical imaging). Fundamental mathematical research is absolutely necessary if Australia is to maintain a presence on the international scientific stage.
Read moreRead less
Classification and Invariants in Complex Differential Geometry. Differential geometry is the study of shape using calculus and differential equations. This is a fundamental research project in this area. Complex differential geometry refers to geometry based on the complex numbers, generally a rich and intriguing setting. Geometries will be distinguished by the construction of suitable invariants, both algebraic and analytic. Classification problems will be solved by these means. Of particular i ....Classification and Invariants in Complex Differential Geometry. Differential geometry is the study of shape using calculus and differential equations. This is a fundamental research project in this area. Complex differential geometry refers to geometry based on the complex numbers, generally a rich and intriguing setting. Geometries will be distinguished by the construction of suitable invariants, both algebraic and analytic. Classification problems will be solved by these means. Of particular interest are geometries with a high degree of symmetry, a critical feature that pervades both mathematics and physics. Twistor theory provides the unifying theme for this project.Read moreRead less
Homotopical structures in algebraic, analytic, and equivariant geometry. This is a project for fundamental research in pure mathematics. It is focused on an emerging subfield of complex geometry concerned with spaces and maps that exhibit exceptional flexibility properties, which often go hand-in-hand with a high degree of symmetry. The project aims to develop the foundations of this new area, solve several open problems, and pursue interconnections with and applications to algebraic geometry, c ....Homotopical structures in algebraic, analytic, and equivariant geometry. This is a project for fundamental research in pure mathematics. It is focused on an emerging subfield of complex geometry concerned with spaces and maps that exhibit exceptional flexibility properties, which often go hand-in-hand with a high degree of symmetry. The project aims to develop the foundations of this new area, solve several open problems, and pursue interconnections with and applications to algebraic geometry, complex analysis, geometric invariant theory, and topology.Read moreRead less
Symmetries in real and complex geometry. This project concerns an important area of abstract modern geometry. The results and techniques of the project will lead to significant progress in this area. It will benefit the national scientific reputation, strengthen the research profile of the home institutions, and provide training to young researchers.
Flexibility and symmetry in complex geometry. Differential equations play a fundamental role in science and technology. The aim of the project is to study important differential equations that arise in geometry, their symmetries, and obstructions to solving them.
Normal forms and Chern-Moser connection in the study of Cauchy-Riemann Manifolds. This research project is aimed at a systematic study of Cauchy-Riemann manifolds, their holomorphic mappings and automorphisms, by means of a unifying approach based on
Chern-Moser type normal forms. The importance of Cauchy-Riemann manifolds stems from the fact that they bridge complex structure and holomorphy with the Riemannian nature of real manifolds. Construction of an analogue of the Chern-Moser normal form ....Normal forms and Chern-Moser connection in the study of Cauchy-Riemann Manifolds. This research project is aimed at a systematic study of Cauchy-Riemann manifolds, their holomorphic mappings and automorphisms, by means of a unifying approach based on
Chern-Moser type normal forms. The importance of Cauchy-Riemann manifolds stems from the fact that they bridge complex structure and holomorphy with the Riemannian nature of real manifolds. Construction of an analogue of the Chern-Moser normal form for multicodimensional Levi-nondegenerate CR-manifolds and extension of CR-mappings between them are major goals in complex analysis. Identification of Chern-Moser chains and equivariant linearisation of isotropy automorphisms are major goals in geometry.Read moreRead less
Identification Power and Instrument Strength in Discrete Outcome Models. This project aims to develop new econometric and statistical techniques to quantify causal effects in treatment models with discrete outcomes. Expected outcomes include a much-needed weak instrument test, a measure for identification strength in partial identification setting, and an instrument-covariate selection procedure for high dimensional discrete models based identification power. The benefits include advanced knowle ....Identification Power and Instrument Strength in Discrete Outcome Models. This project aims to develop new econometric and statistical techniques to quantify causal effects in treatment models with discrete outcomes. Expected outcomes include a much-needed weak instrument test, a measure for identification strength in partial identification setting, and an instrument-covariate selection procedure for high dimensional discrete models based identification power. The benefits include advanced knowledge in econometrics and statistics, and enhanced tools for program evaluation and policy assessment in empirical causal analysis using observational data. The project falls into the category of smarter information use and is relevant to any national priority areas where policy interventions require assessment.Read moreRead less
Selection of mixed strength moment restrictions and optimal inference . This project aims to develop consistent model selection criteria even if the target model only provides a weak signal about the parameter of interest. This project expects to generate new knowledge on model selection using new and innovative techniques. Expected outcomes include the quantification of the maximum information on parameter from weak-signal models; new entropy-based model selection criteria; and a robust investi ....Selection of mixed strength moment restrictions and optimal inference . This project aims to develop consistent model selection criteria even if the target model only provides a weak signal about the parameter of interest. This project expects to generate new knowledge on model selection using new and innovative techniques. Expected outcomes include the quantification of the maximum information on parameter from weak-signal models; new entropy-based model selection criteria; and a robust investigation of the still debated hypothesis in environmental economics that with open and liberalized trade, developing countries would become pollution havens for dirty industries of advanced countries. Success in this undertaking will dramatically enlarge the pool of applied work involving economic models with weak signals.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE160101565
Funder
Australian Research Council
Funding Amount
$330,000.00
Summary
Flexible data modelling via skew mixture models:challenges and applications. This project seeks to explore new models for handling data with non-normal features. Parametric distributions are fundamental to statistical modelling and inference. For centuries, the ‘normal’ distribution has been the dominant model for continuous data. However, real data rarely satisfy the assumption of normality. There is thus a strong demand for more flexible distributions. This project aims to develop new methodol ....Flexible data modelling via skew mixture models:challenges and applications. This project seeks to explore new models for handling data with non-normal features. Parametric distributions are fundamental to statistical modelling and inference. For centuries, the ‘normal’ distribution has been the dominant model for continuous data. However, real data rarely satisfy the assumption of normality. There is thus a strong demand for more flexible distributions. This project aims to develop new methodologies in finite mixture modelling using skew component distributions to provide better models for handling data with non-normal features (such as skewness, heavy/light tails, and multimodality). Applications may include security intrusion detection, clinical diagnosis and prognosis, and flow and mass cytometry.Read moreRead less
Literacy Instruction for Children with Autism. There is a social and economic imperative to assist all individuals to reach their full potential with regard to literacy skills. The central goal of the proposed research is to develop and evaluate new ways to support comprehensive literacy instruction for children with autism spectrum disorder (ASD). Following our world-first pilot research, we will investigate the efficacy of the ABRACADABRA literacy instruction program delivered in small groups ....Literacy Instruction for Children with Autism. There is a social and economic imperative to assist all individuals to reach their full potential with regard to literacy skills. The central goal of the proposed research is to develop and evaluate new ways to support comprehensive literacy instruction for children with autism spectrum disorder (ASD). Following our world-first pilot research, we will investigate the efficacy of the ABRACADABRA literacy instruction program delivered in small groups of children with ASD supplemented by a novel parent-child shared book reading program. Immediate outcomes extend beyond advances in the science of reading to upskilling parents and professionals who support children with ASD, and provision of tangible life-long benefits for individuals with ASD.Read moreRead less