Occupational Measures Approach to Long Run Average and Singularly Perturbed Optimal Control Problems. Problems of optimal control of long-run average and singularly perturbed systems arise in many applications. The project will lead to the development of new linear programming based techniques for analyzing these problems (including problems intractable so far) and finding their numerical solutions. The new techniques will have a potential to be further developed into software that can benefit A ....Occupational Measures Approach to Long Run Average and Singularly Perturbed Optimal Control Problems. Problems of optimal control of long-run average and singularly perturbed systems arise in many applications. The project will lead to the development of new linear programming based techniques for analyzing these problems (including problems intractable so far) and finding their numerical solutions. The new techniques will have a potential to be further developed into software that can benefit Australian industries and technologies. The proposed topic is in the focus of interest of many eminent researchers around the world and the dissemination of our results will further improve Australia's standing in the international research community. Read moreRead less
Duality, singular perturbations and numerical analysis in infinite dimensional linear programming problems related to problems of control of nonlinear dynamical systems. Problems of control of nonlinear dynamical systems attract continued interest of eminent researchers motivated by important applications and by the fact that analytical and/or numerical analysis of a general nonlinear control problem presents a challenging task. The outcomes of the project will be both fundamental theoretical re ....Duality, singular perturbations and numerical analysis in infinite dimensional linear programming problems related to problems of control of nonlinear dynamical systems. Problems of control of nonlinear dynamical systems attract continued interest of eminent researchers motivated by important applications and by the fact that analytical and/or numerical analysis of a general nonlinear control problem presents a challenging task. The outcomes of the project will be both fundamental theoretical results and readily applicable (linear programming based) algorithms that will equip researchers and engineers with new tools for analysis and numerical solution of nonlinear control problems (including problems that have been intractable so far). The project will further enhance Australia's international reputation in Control Theory and its Applications.
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Linear programming approach to nonlinear deterministic and stochastic control problems: perturbations methods and numerical analysis. The proposed research will significantly advance knowledge by creating new analytical and numerical methods for tackling complex nonlinear control problems arising in many applications. The study's outputs will lead to a deeper understanding of fundamental issues in mathematical modelling. Collaboration with renowned researchers will further improve Australia's st ....Linear programming approach to nonlinear deterministic and stochastic control problems: perturbations methods and numerical analysis. The proposed research will significantly advance knowledge by creating new analytical and numerical methods for tackling complex nonlinear control problems arising in many applications. The study's outputs will lead to a deeper understanding of fundamental issues in mathematical modelling. Collaboration with renowned researchers will further improve Australia's standing in the international research community. Also their visits may further promote research both within and outside the host institution. In particular, lectures and seminars that they will deliver will be transmitted to Australian universities participating in the Access Grid Room Project.Read moreRead less
Higher Line Bundles in Geometry and Physics. This project seeks to develop a theory of geometric objects, `higher line bundles', which realise elements of higher dimensional cohomology groups. In particular this project will develop a theory of differential geometry for these objects, allowing one to interpret differential forms representing cohomology classes as the `curvature' of a higher line bundle. This will have applications in quantum field theory and string/brane theory.
Global aspects of dualities in String Theory in the presence of background fluxes. String Theory, known to the general public as the "Theory of Everything', is currently an extremely active area of research internationally. It has not only stimulated considerable interaction between mathematical physicists and mathematicians, but also increased public interest in science through television programs and books. Unfortunately, the majority of the Australian scientific community has not yet caught ....Global aspects of dualities in String Theory in the presence of background fluxes. String Theory, known to the general public as the "Theory of Everything', is currently an extremely active area of research internationally. It has not only stimulated considerable interaction between mathematical physicists and mathematicians, but also increased public interest in science through television programs and books. Unfortunately, the majority of the Australian scientific community has not yet caught up with these developments. Our recent papers, all published in premier journals in this field, have not only received widespread international attention but have also increased the profile of String Theory amongst Australia's mathematicians and mathematical physicists. The proposed project is expected to continue this trend.Read moreRead less
Twisted K-theory and its application to String Theory and Conformal Field Theory. String Theory is, at present, the only consistent theory of quantum gravity. Recently, twisted K-theory was proposed as the algebraic structure underlying the classification of D-branes, i.e. solitonic extended objects, in certain closed string backgrounds. In this project we aim to advance our understanding of the properties of twisted K-theory in the context of String Theory and Conformal Field Theory. The ult ....Twisted K-theory and its application to String Theory and Conformal Field Theory. String Theory is, at present, the only consistent theory of quantum gravity. Recently, twisted K-theory was proposed as the algebraic structure underlying the classification of D-branes, i.e. solitonic extended objects, in certain closed string backgrounds. In this project we aim to advance our understanding of the properties of twisted K-theory in the context of String Theory and Conformal Field Theory. The ultimate goal is to find the appropriate K-theory classifying D-branes in arbitrary closed string backgrounds or, similarly, classifying boundary Conformal Field Theories. It has already emerged that the K-theory of C*-algebras will play an important role.Read moreRead less
Dualities in String Theory and Conformal Field Theory in the context of the Geometric Langlands Program. The Langlands program ties together seemingly unrelated areas of Mathematics. Recently, in the context of the Geometric Langlands correspondence, novel connections with Theoretical Physics have emerged, thus becoming one of the most active areas of research in both Mathematics and Theoretical Physics. Australia has a number of world-renowned experts, including the two CI's, in various aspect ....Dualities in String Theory and Conformal Field Theory in the context of the Geometric Langlands Program. The Langlands program ties together seemingly unrelated areas of Mathematics. Recently, in the context of the Geometric Langlands correspondence, novel connections with Theoretical Physics have emerged, thus becoming one of the most active areas of research in both Mathematics and Theoretical Physics. Australia has a number of world-renowned experts, including the two CI's, in various aspects of the Langlands program, and is therefore well-placed to make seminal contributions. Being involved in these new developments is of crucial importance to the health of Mathematics and Theoretical Physics in Australia. An integral part of this proposal is student involvement and postgraduate training.Read moreRead less
Fractional analytic index theory. Atiyah-Singer index theory, for which M.F. Atiyah and I.M. Singer received the 2004 Abel Prize, has stimulated considerable interaction between mathematicians and mathematical physicists. An extension of this theory is Fractional Index Theory, co-invented by R.B. Melrose, I.M. Singer and myself, which has received international attention, having solved a fundamental open problem. A central aim in my research project is to extend our theory to elliptic boundary ....Fractional analytic index theory. Atiyah-Singer index theory, for which M.F. Atiyah and I.M. Singer received the 2004 Abel Prize, has stimulated considerable interaction between mathematicians and mathematical physicists. An extension of this theory is Fractional Index Theory, co-invented by R.B. Melrose, I.M. Singer and myself, which has received international attention, having solved a fundamental open problem. A central aim in my research project is to extend our theory to elliptic boundary value problems. I will assist beginners to navigate to the cutting edge of research through workshops, spring-schools and supervision. Benefits include the enhancement of Australia's position in the forefront of international research.Read moreRead less
New perspectives on computing methods for mathematical signal processing. This project determines how best to design computing methods for challenging demands in signal processing. The expected conceptual & algorithmic advances will have significant repercussions in a number of fields including optimal filtering theory and will contribute to applications ranging from bio-informatics to electrical engineering. The new techniques will allow development of software that will benefit Australian in ....New perspectives on computing methods for mathematical signal processing. This project determines how best to design computing methods for challenging demands in signal processing. The expected conceptual & algorithmic advances will have significant repercussions in a number of fields including optimal filtering theory and will contribute to applications ranging from bio-informatics to electrical engineering. The new techniques will allow development of software that will benefit Australian industries and technologies. The formation of a strong research team across four universities in Australia, USA and Japan will enhance our scientific standing in the international community and will place Australian researchers at the forefront of world-class research methods. Read moreRead less
Modelling mean and dispersion using fixed and random effects. The aims of the project are to develop methods for joint mean and dispersion modelling using fixed and random effects, in the generalized linear models context and for Gaussian longitudinal data. The significance is the more efficient, precise and appropriate analysis of data arising from many areas of application. The expected outcomes are therefore better methods of analysis, software to carry the analyses out, and potentially impor ....Modelling mean and dispersion using fixed and random effects. The aims of the project are to develop methods for joint mean and dispersion modelling using fixed and random effects, in the generalized linear models context and for Gaussian longitudinal data. The significance is the more efficient, precise and appropriate analysis of data arising from many areas of application. The expected outcomes are therefore better methods of analysis, software to carry the analyses out, and potentially important results in applications.Read moreRead less