Group orbits in garmonic analysis and ergodic theory. Researchers from many areas need a type of mathematical analysis which involves the behaviour of a system - which may be a set of data points - under repeated application of some operation or group of operations. The structures arising from this kind of process are known as group orbits. The project gives information about their nature. Two major types of orbits are considered, coming from actions of discrete groups on measure spaces, and fro ....Group orbits in garmonic analysis and ergodic theory. Researchers from many areas need a type of mathematical analysis which involves the behaviour of a system - which may be a set of data points - under repeated application of some operation or group of operations. The structures arising from this kind of process are known as group orbits. The project gives information about their nature. Two major types of orbits are considered, coming from actions of discrete groups on measure spaces, and from smooth actions of Lie groups on manifolds, where powerful geometric methods are available. The project will yield new understandings of entropy, and new approaches to Fourier analysis.Read moreRead less
Ergodic theory and number theory. Recent advances in the theory of measured dynamical systems investigated by the proponents include new versions of entropy, and the study of spectral theory for non-singular systems. These will be further developed in this joint project with the French CNRS. The results are expected to have interesting applications in physics and number theory.
Operator Integrals and Derivatives. The project is a contribution to the study of non-commutative differential and integral calculus. The novelty of the present project lies in the study of smoothness properties of functions whose domains and ranges are spaces of unbounded, non-commuting operators on some Hilbert space. Our general approach will be based on a detailed investigation of properties of double operator integrals, which permit smoothness estimates of operator-functions. It can be expe ....Operator Integrals and Derivatives. The project is a contribution to the study of non-commutative differential and integral calculus. The novelty of the present project lies in the study of smoothness properties of functions whose domains and ranges are spaces of unbounded, non-commuting operators on some Hilbert space. Our general approach will be based on a detailed investigation of properties of double operator integrals, which permit smoothness estimates of operator-functions. It can be expected that the new techniques generated will find further application in areas of mathematical physics and non-commutative geometry related to quantized calculus.
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Entropy and maximal entropy in Markov systems. Entropy is a measure of how well-ordered a system is: chaotic systems have high entropy. Two approaches to entropy are available, via the limiting behaviour of the orbits of points, which yields topological entropy, and via the behaviour of the distributions of measures of partitions, yielding measure-theoretic entropy. The topological entropy is the least upper bound of entropies of all possible measures. We study when there is a measure which real ....Entropy and maximal entropy in Markov systems. Entropy is a measure of how well-ordered a system is: chaotic systems have high entropy. Two approaches to entropy are available, via the limiting behaviour of the orbits of points, which yields topological entropy, and via the behaviour of the distributions of measures of partitions, yielding measure-theoretic entropy. The topological entropy is the least upper bound of entropies of all possible measures. We study when there is a measure which realises this bound, describing the structure of such systems via Markov and Bratteli diagrams. Our methods will be applied to new versions of entropy for non-singular systems. This will assist in the description of chaotic behaviour.Read moreRead less
Symmetries in analysis. Technical research is like an iceberg. The 10% you see in applications is supported by 90% hidden, long-term, sometimes abstruse or theoretical-sounding work. The area of mathematical analysis has, for over 200 years, proved its worth as part of the unseen 90%, giving us such important tools as Fourier analysis, statistical mechanics and quantum mechanics. Australia is known as a world leader in mathematical analysis, and it is important for the country to maintain that e ....Symmetries in analysis. Technical research is like an iceberg. The 10% you see in applications is supported by 90% hidden, long-term, sometimes abstruse or theoretical-sounding work. The area of mathematical analysis has, for over 200 years, proved its worth as part of the unseen 90%, giving us such important tools as Fourier analysis, statistical mechanics and quantum mechanics. Australia is known as a world leader in mathematical analysis, and it is important for the country to maintain that edge in a number of key disciplines, so we can continue to participate in global technological advance. The project has an international focus which will enable that to happen. It will also provide training for the next generation of mathematicians. Read moreRead less
Dynamical systems: theory and practice. Mathematical science has proven a crucial platform for science and technology: it may have a long lead-time to application but its impacts are more profound than glamorous technical developments. Australia has an economic imperative to maintain investment in fundamental mathematics. Dynamical systems underpin a wide range of applications in physics, engineering, information science, finance and economics. This project will improve our capacity to model sy ....Dynamical systems: theory and practice. Mathematical science has proven a crucial platform for science and technology: it may have a long lead-time to application but its impacts are more profound than glamorous technical developments. Australia has an economic imperative to maintain investment in fundamental mathematics. Dynamical systems underpin a wide range of applications in physics, engineering, information science, finance and economics. This project will improve our capacity to model systems and to study their evolution, giving us better predictive power. It will keep Australia in the forefront of international research, providing a basis of expertise not otherwise available to Australian researchers and industry. Read moreRead less
Non-commutative analysis and differential calculus. This project is in an area of central mathematical importance and will lead to important scientific advances that will keep Australia at the forefront internationally in this field of research. There is an emphasis on international networking and we will collaborate with leading researchers in USA and France.
Use Of Expression Profiling To Identify Genes Influencing Cardiovascular Risk In The Norfolk Island Population Isolate
Funder
National Health and Medical Research Council
Funding Amount
$697,409.00
Summary
This study will use a unique population isolate from Norfolk Island. We aim to identify genes that play a role in cardiovascular disease risk. Norfolk has a population of ~1200 permanent residents, most of whom are direct descendents of 18th century English Bounty mutineers and Polynesian women. We will undertake gene expression mapping to identify genomic loci that influence cardiovascular disease using samples from this population isolate.
Fine Scale Mapping And Identification Of The IBD1 Gene On Chromsosome 16
Funder
National Health and Medical Research Council
Funding Amount
$483,849.00
Summary
One of the greatest challenges facing contemporary gastroenterology is to understand the causes of the inflammatory bowel diseases (IBD). Studies on the prevalence, incidence and cost of IBD indicate that these diseases have considerable impact in Australia. On average, patients lose more than 13 days from work each year, and in hospital, IBD in-patients accounted for 7% of total admissions and 10% of total bed days at an average cost of $2600 per admission. We estimate that there may be more th ....One of the greatest challenges facing contemporary gastroenterology is to understand the causes of the inflammatory bowel diseases (IBD). Studies on the prevalence, incidence and cost of IBD indicate that these diseases have considerable impact in Australia. On average, patients lose more than 13 days from work each year, and in hospital, IBD in-patients accounted for 7% of total admissions and 10% of total bed days at an average cost of $2600 per admission. We estimate that there may be more than 10,000 Australians who suffer from IBD. The existence of a genetic predisposition to IBD is now well established, and there is strong evidence that the disease is complex, resulting from the interaction of a number of different genes. To date, one genetic localisation on chromosome 16 has been established in several different populations, and we have confirmed the importance of this localisation in the Australian population. We will further refine the localisation by fine scale mapping in the pericentromeric region of chromosome 16 by identifying and studying the inheritance of novel markers in the region. We will then identify and characterise the gene itself using several complementary appoaches that rely on differences at the molecular level between disease and normal tissue. This work is part of the international effort to identify all IBD susceptibility genes. Once that is achieved, approaches to explaining the interactions between the genes, their protein products and environmental triggers can be determined. Only when the mechanisms of these interactions are understood will the expectation of rational therapies based on an understanding of disease aetiology be possible.Read moreRead less