Exploring the Frontiers of Feasible Computation. The project aims to delineate the boundary between feasible and infeasible computational problems. A problem is considered feasible if there is an algorithm to solve it in worst-case time bounded by a polynomial in the input size. This is probably impossible for the important class of NP-complete problems. However, typical examples of NP-complete problems can often be solved in polynomial time, because worst-case problems are rare. The project is ....Exploring the Frontiers of Feasible Computation. The project aims to delineate the boundary between feasible and infeasible computational problems. A problem is considered feasible if there is an algorithm to solve it in worst-case time bounded by a polynomial in the input size. This is probably impossible for the important class of NP-complete problems. However, typical examples of NP-complete problems can often be solved in polynomial time, because worst-case problems are rare. The project is relevant to public-key cryptography, where breaking an encryption scheme should be infeasible, and to many real-life situations where NP-complete problems need to be solved, either exactly or approximately.Read moreRead less
Multicast in Single-Hop and Multi-Hop WDM Optical Networks. The emerging Wavelength-Division-Multiplexing (WDM) optical network is a promising candidate for next-generation Internet, which provides enormous bandwidth and fast connectivity. Multicast in WDM networks is a fundamental problem which has wide applications including teleconferencing, entertainment distribution, etc. In this project we investigate the multicast and constraint multicast problems in both single-hop and multi-hop WDM netw ....Multicast in Single-Hop and Multi-Hop WDM Optical Networks. The emerging Wavelength-Division-Multiplexing (WDM) optical network is a promising candidate for next-generation Internet, which provides enormous bandwidth and fast connectivity. Multicast in WDM networks is a fundamental problem which has wide applications including teleconferencing, entertainment distribution, etc. In this project we investigate the multicast and constraint multicast problems in both single-hop and multi-hop WDM networks by studying their computational complexities and devising scalable, high-quality approximation algorithms for them. The developed algorithms significantly improve the network performance and scalability, and the innovative approaches and algorithm techniques developed in this project are also applicable to other routing problems.Read moreRead less
Expressive power and complexity of temporal logics for model-checking. Hardware verification based upon mathematical logic is now routinely
used in industry to verify the correctness of large digital circuits
using a technique called model-checking. Such discrete systems move
from one state to another according to the regular ticks of a clock.
The challenge now is to find tractable methods for reasoning about
real-time systems and hybrid systems that move in a continuous manner
with respec ....Expressive power and complexity of temporal logics for model-checking. Hardware verification based upon mathematical logic is now routinely
used in industry to verify the correctness of large digital circuits
using a technique called model-checking. Such discrete systems move
from one state to another according to the regular ticks of a clock.
The challenge now is to find tractable methods for reasoning about
real-time systems and hybrid systems that move in a continuous manner
with respect to time: examples include aeroplanes flying according to
the laws of physics and a moving robot arm. We shall invent new logics
which are specifically tailored for tractable reasoning about
real-time and hybrid systems.Read moreRead less
HARMONIC ANALYSIS AND BOUNDARY VALUE PROBLEMS FOR ELLIPTIC SYSTEMS. It is of the utmost necessity for Australia to develop the theoretical
expertise needed in the current era. The type of mathematics under
investigation here is closely allied to that needed in much of the
current boom in communication technology and medical research. The
training which would be provided to the research associates is
considerable, and would flow on to produce the expertise needed to
keep the coming gen ....HARMONIC ANALYSIS AND BOUNDARY VALUE PROBLEMS FOR ELLIPTIC SYSTEMS. It is of the utmost necessity for Australia to develop the theoretical
expertise needed in the current era. The type of mathematics under
investigation here is closely allied to that needed in much of the
current boom in communication technology and medical research. The
training which would be provided to the research associates is
considerable, and would flow on to produce the expertise needed to
keep the coming generation involved in modern technological development. I will maintain my large collaborative effort with
leading mathematicians from the US, France and other countries, thus
helping to keep Australia at the forefront of a significant field of
research.Read moreRead less
HARMONIC ANALYSIS OF ELLIPTIC SYSTEMS ON RIEMANNIAN MANIFOLDS. It is of the utmost necessity for Australia to develop the theoretical expertise needed in the current era. The type of mathematics under investigation here is closely allied to that needed in much of the current boom in communication technology and medical research. The training which would be provided to the research associates is considerable, and would flow on to produce the expertise needed to keep the coming generation invol ....HARMONIC ANALYSIS OF ELLIPTIC SYSTEMS ON RIEMANNIAN MANIFOLDS. It is of the utmost necessity for Australia to develop the theoretical expertise needed in the current era. The type of mathematics under investigation here is closely allied to that needed in much of the current boom in communication technology and medical research. The training which would be provided to the research associates is considerable, and would flow on to produce the expertise needed to keep the coming generation involved in modern technological development. I will maintain my active collaborative effort with leading mathematicians from the US, France and other countries, thus helping to keep Australia at the forefront of a significant field of research.Read moreRead less
Propagation of singularities for the Schrodinger equation. The time-dependent Schrodinger equation governs the evolution of quantum particles. In this project we aim to use new techniques from mathematical scattering theory to analyse solutions of the Schrodinger equation and obtain sharp bounds on their singularities. Controlling such singularities will allow us to deduce quantitative bounds on the number of eigenvalues in certain situations, and provide new techniques for studying nonlinear Sc ....Propagation of singularities for the Schrodinger equation. The time-dependent Schrodinger equation governs the evolution of quantum particles. In this project we aim to use new techniques from mathematical scattering theory to analyse solutions of the Schrodinger equation and obtain sharp bounds on their singularities. Controlling such singularities will allow us to deduce quantitative bounds on the number of eigenvalues in certain situations, and provide new techniques for studying nonlinear Schrodinger equations. Read moreRead less
Physics of Risk: new tools to survey the Australian market and beyond. The lives of most Australians depend on the dynamics of financial markets that affects investments, savings, business, employment, growth, wealth and -ultimately- the daily functioning of our society. Understanding, monitoring and managing the dynamics of financial markets is of crucial importance to policy-makers, financial institutions and businesses that are increasingly faced with managing risk, planning strategies and ta ....Physics of Risk: new tools to survey the Australian market and beyond. The lives of most Australians depend on the dynamics of financial markets that affects investments, savings, business, employment, growth, wealth and -ultimately- the daily functioning of our society. Understanding, monitoring and managing the dynamics of financial markets is of crucial importance to policy-makers, financial institutions and businesses that are increasingly faced with managing risk, planning strategies and taking decisions in an increasingly complex market-place. The project is also of importance to the continued evolution of physics in this country contributing to the emergence of a strong new area of statistical physics concerned with the ?real world? in a manner hitherto unknown.Read moreRead less
HARMONIC ANALYSIS, BOUNDARY VALUE PROBLEMS, AND MAXWELL'S EQUATIONS IN LIPSCHITZ DOMAINS. Boundary value problems for partial differential equations arise naturally when physical problems are expressed in mathematical terms. This project concerns the systematic development of the harmonic analysis of partial differential operators, and of the corresponding boundary integrals in order to solve such problems on irregular regions. Particular emphasis is given to studying the behaviour of electrom ....HARMONIC ANALYSIS, BOUNDARY VALUE PROBLEMS, AND MAXWELL'S EQUATIONS IN LIPSCHITZ DOMAINS. Boundary value problems for partial differential equations arise naturally when physical problems are expressed in mathematical terms. This project concerns the systematic development of the harmonic analysis of partial differential operators, and of the corresponding boundary integrals in order to solve such problems on irregular regions. Particular emphasis is given to studying the behaviour of electromagnetic waves both inside and outside irregularly shaped surfaces, and their propagation through it.Read moreRead less
Large Scale Complex Multiagent Systems : Control Methodologies and Information Architectures. Future military operations are likely to involve increasing use of unmanned vehicles, airborne, underwater or on land. In some cases there will be very large formations. This research will develop methodologies which allow formations of unmanned vehicles to be configured. Much of the same methodology is applicable to the construction and operation of large scale sensor networks, identified by some comm ....Large Scale Complex Multiagent Systems : Control Methodologies and Information Architectures. Future military operations are likely to involve increasing use of unmanned vehicles, airborne, underwater or on land. In some cases there will be very large formations. This research will develop methodologies which allow formations of unmanned vehicles to be configured. Much of the same methodology is applicable to the construction and operation of large scale sensor networks, identified by some commentators as one of the most important technologies of the 21st century. They comprise large numbers of low cost networked sensors and will increasingly find application in security, agricultural and environmental monitoring.Read moreRead less
Development of robust adaptive and nonlinear control methodologies. Techniques will be developed that will allow industrial plants to operate with lower direct or environmental cost, and airborne vehicles to operate more efficiently or more safely.