Stein's method for probability approximation. Data of counts in time, such as incoming calls in telecommunications and the clusters of palindromes in a family of herpes-virus genomes, arise in an extraordinarily diverse range of fields from science to business. These problems can be modelled by sums of random variables taking values 0 and 1 in probability theory, thus permitting approximate calculations which are often good enough in practice. This project will obtain such approximate solutions ....Stein's method for probability approximation. Data of counts in time, such as incoming calls in telecommunications and the clusters of palindromes in a family of herpes-virus genomes, arise in an extraordinarily diverse range of fields from science to business. These problems can be modelled by sums of random variables taking values 0 and 1 in probability theory, thus permitting approximate calculations which are often good enough in practice. This project will obtain such approximate solutions and estimate the errors involved. Applications include analysis of data in insurance, finance, flood prediction in hydrology.Read moreRead less
Random network models with applications in biology. Complex biological systems consist of a large number of interacting agents or components, and so can be studied using mathematical random network models. We aim to gain deeper insights into the laws emerging as the random networks evolve in time. This can help us to deal with dangerous disease epidemics and better understand the human brain.
Random Discrete Structures: Approximations and Applications. The behaviour of many real world systems can be modelled by random discrete structures evolving over time. For example, the sizes of populations of frogs in some close patches of forests can be modelled as interacting random processes. The aim of the project is to investigate large discrete random structures that arise from real world application in areas such as biology, complex networks and insurance. The proposed project is at the i ....Random Discrete Structures: Approximations and Applications. The behaviour of many real world systems can be modelled by random discrete structures evolving over time. For example, the sizes of populations of frogs in some close patches of forests can be modelled as interacting random processes. The aim of the project is to investigate large discrete random structures that arise from real world application in areas such as biology, complex networks and insurance. The proposed project is at the interface of mathematics and 'big data' applications and so the work of the project aims to provide theoretical and heuristic underpinnings useful in the algorithms and techniques of practitioners. Understanding the applications in the project requires new, broadly applicable methods and developing such is a complementary aim.Read moreRead less
Quantum decoherence: A game-theoretic perspective. Algorithms based on quantum computation have the ability to significantly speed up information processing compared to standard computers. The increase in computational power can have enormous impact on humankind and this project will help maintain Australia's position in the global forefront of this effort.This project focuses on the thoeretical foundations of quantum computation and complements the efforts of several groups in Australia collabo ....Quantum decoherence: A game-theoretic perspective. Algorithms based on quantum computation have the ability to significantly speed up information processing compared to standard computers. The increase in computational power can have enormous impact on humankind and this project will help maintain Australia's position in the global forefront of this effort.This project focuses on the thoeretical foundations of quantum computation and complements the efforts of several groups in Australia collaborating on the experimental design of quantum computers. The project will increase the fundamental understanding of how quantum information is processed in the presence of noise, which is necessary for the successful operation of quantum computers. Read moreRead less
Asymptotic Expansions and Large Deviations in Probability and Statistics: Theory and Applications. Statistics is the major enabling science in a number of disciplines. This is fundamental research in probability and statistics but it has wide applications in Biology and Social Sciences which will ultimately be of national benefit. The behaviour of self normalized sums is an exciting new area of fundamental research that has implications for the application of statistics in many areas. U-statist ....Asymptotic Expansions and Large Deviations in Probability and Statistics: Theory and Applications. Statistics is the major enabling science in a number of disciplines. This is fundamental research in probability and statistics but it has wide applications in Biology and Social Sciences which will ultimately be of national benefit. The behaviour of self normalized sums is an exciting new area of fundamental research that has implications for the application of statistics in many areas. U-statistics for dependent situations has direct application to understanding financial time series and the analysis of sample survey data. Saddlepoint methods provide extremely accurate approximations in a number of important applications.
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Financial Risk Processes: Stochastic and Statistical Models and their Applications. On the one hand, the misuse of complex financial instruments has contributed to recent major disasters in the Australian financial and insurance industries; on the other hand, great benefits can be obtained by correct use of these kinds of instruments, to share risk between markets and segments of markets. The overall research effort in Australia in these areas is relatively small. This project will target the de ....Financial Risk Processes: Stochastic and Statistical Models and their Applications. On the one hand, the misuse of complex financial instruments has contributed to recent major disasters in the Australian financial and insurance industries; on the other hand, great benefits can be obtained by correct use of these kinds of instruments, to share risk between markets and segments of markets. The overall research effort in Australia in these areas is relatively small. This project will target the development of cutting edge technologies underlying the use of financial derivatives, not presently studied in this country or elsewhere, by bringing together a variety of top level international researchers in an integrated effort to lift the Australian understanding and application of this methodology.Read moreRead less
Markov Field Theory applied to Sensor Networks Analysis and Design. Ad hoc and sensor networks have a wide range of applications in defence, emergency services and agriculture because they do not require telecommunications infrastructure such as base stations or access points, hence are relatively easy to deploy in harsh environments. This project aims at improving the theoretical understanding of sensor and ad hoc networks, which enable improvements in performance in such networks. Australian d ....Markov Field Theory applied to Sensor Networks Analysis and Design. Ad hoc and sensor networks have a wide range of applications in defence, emergency services and agriculture because they do not require telecommunications infrastructure such as base stations or access points, hence are relatively easy to deploy in harsh environments. This project aims at improving the theoretical understanding of sensor and ad hoc networks, which enable improvements in performance in such networks. Australian defence industry and emergency services will benefit from this research by gaining access to improved ad hoc communications networks. The agricultural sector will also benefit from the improved sensor networks in applications such as monitoring soil conditions, stock and crop levels. Read moreRead less
Hypergraph models for complex discrete systems. This project aims to better understand the structure and properties of very large hypergraphs of various kinds. Hypergraphs are very general mathematical objects which can be used to model complex discrete systems. They arise naturally in many areas such as ecology, chemistry and computer science. Despite this, our theoretical understanding of very large, or random, hypergraphs lags far behind the intensely-studied special case of graphs. This proj ....Hypergraph models for complex discrete systems. This project aims to better understand the structure and properties of very large hypergraphs of various kinds. Hypergraphs are very general mathematical objects which can be used to model complex discrete systems. They arise naturally in many areas such as ecology, chemistry and computer science. Despite this, our theoretical understanding of very large, or random, hypergraphs lags far behind the intensely-studied special case of graphs. This project will answer many fundamental questions about large, random hypergraphs. The expected outcomes of the project also include new tools for working with hypergraphs, such as efficient algorithms for sampling hypergraphs. These outcomes will benefit researchers who use hypergraphs in their work and will enhance Australia's reputation for research in this area.Read moreRead less
Random walks with long memory. This project aims to study novel random walk models with long memory, including systems of multiple random walkers that interact through their environment. This would provide a mathematical understanding of phenomena such as aggregation in colonies of bacteria, and ant colony optimisation algorithms. The project aims to produce highly cited publications, and to train future researchers.
New universality in stochastic systems. This project aims to uncover new analyses and effects in the complex behaviour of non-linear systems with random noise. Many systems originate near an unstable equilibrium. This project will develop a new mathematical theory that establishes a universality in the way the long term effect of noise expresses itself as random initial conditions in the dynamics. It will fill gaps in Mathematics and make refinements to existing fundamental scientific laws by in ....New universality in stochastic systems. This project aims to uncover new analyses and effects in the complex behaviour of non-linear systems with random noise. Many systems originate near an unstable equilibrium. This project will develop a new mathematical theory that establishes a universality in the way the long term effect of noise expresses itself as random initial conditions in the dynamics. It will fill gaps in Mathematics and make refinements to existing fundamental scientific laws by including random initial conditions as predicted by our theory. This will advance our understanding of complex systems subjected to noise and will provide significant benefits in the scientific discoveries in Biology, Ecology, Physics and other Sciences where such systems are frequently met.Read moreRead less