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
The impact of crime on the mental wellbeing of communities. Considering not only the direct impact of crime on the victims but also the indirect consequences of living in a community with a higher crime rate may in fact lead to a more accurate analysis of the size of the consequences of criminal activities on the society. Further, it may also be the case that the negative externality of crime on non-victims is much larger for some crimes than for others. If this were the case, it could have impl ....The impact of crime on the mental wellbeing of communities. Considering not only the direct impact of crime on the victims but also the indirect consequences of living in a community with a higher crime rate may in fact lead to a more accurate analysis of the size of the consequences of criminal activities on the society. Further, it may also be the case that the negative externality of crime on non-victims is much larger for some crimes than for others. If this were the case, it could have implications for the way in which police resources are presently distributed across different crimes. Better understanding the total societal cost of crime - to both victims and non-victims - could therefore help improve public policy. Read moreRead less
Hardy spaces of differential forms and applications. Hardy spaces on Euclidean spaces were developed in the 1970's following the fundamental work of Stein, Weiss and Fefferman. These spaces play an important role in harmonic analysis, as they are the natural spaces on which to consider singular integral operators. They arise in many contexts, such as when using Jacobians in non-linear partial differential equations. Recently the French participants and the Australian participants have have obt ....Hardy spaces of differential forms and applications. Hardy spaces on Euclidean spaces were developed in the 1970's following the fundamental work of Stein, Weiss and Fefferman. These spaces play an important role in harmonic analysis, as they are the natural spaces on which to consider singular integral operators. They arise in many contexts, such as when using Jacobians in non-linear partial differential equations. Recently the French participants and the Australian participants have have obtained different but related results concerning Hardy spaces of exact differential forms. The time is now ripe to construct a unified theory.
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Asymptotic Geometric Analysis and Machine Learning. Phenomena in large dimensions appear in a number of domains of Mathematics and adjacent domains of science (e.g. Computer Science), dealing with functions of infinitely growing number of parameters. Here, we focus on several questions naturally linked to Asymptotic Geometric Analysis which have natural applications to Statistical Learning Theory. We intend to use geometric, probabilistic and combinatorial methods to investigate these problems, ....Asymptotic Geometric Analysis and Machine Learning. Phenomena in large dimensions appear in a number of domains of Mathematics and adjacent domains of science (e.g. Computer Science), dealing with functions of infinitely growing number of parameters. Here, we focus on several questions naturally linked to Asymptotic Geometric Analysis which have natural applications to Statistical Learning Theory. We intend to use geometric, probabilistic and combinatorial methods to investigate these problems, with an emphasis on modern tools in Empirical Processes Theory and the theory of Random Matrices.Read moreRead less
International linkages between financial and real economy cycles. This project will develop empirical multivariate models of business cycles for different countries, and will study how they are linked via international financial markets. The significance of the project is that the models will integrate previous work done on financial-real economy links in single country settings, and it will explicitly study the effects of disequilibrium in international financial markets (such as overvaluation ....International linkages between financial and real economy cycles. This project will develop empirical multivariate models of business cycles for different countries, and will study how they are linked via international financial markets. The significance of the project is that the models will integrate previous work done on financial-real economy links in single country settings, and it will explicitly study the effects of disequilibrium in international financial markets (such as overvaluation of foreign currency) on individual economies. The project will determine the appropriate domestic monetary policy response to such shocks, and the potential for forecasting both the advent of such shocks and their impact on individual economies.Read moreRead less
What are the determinants and effects of school choice? An international comparison between Australia and the UK. The project will provide a clearer picture of the determinants of school choice, most notably the impact of economic phenomena on that choice. For Australia, this will mean we have a better understanding of the competing explanations of the growth of students at private schools over the past three decades: the role of government subsidies; the role of demographic changes; the effec ....What are the determinants and effects of school choice? An international comparison between Australia and the UK. The project will provide a clearer picture of the determinants of school choice, most notably the impact of economic phenomena on that choice. For Australia, this will mean we have a better understanding of the competing explanations of the growth of students at private schools over the past three decades: the role of government subsidies; the role of demographic changes; the effects of increased income inequality; and the role of changes in preferences towards private schools. This information will provide a better evidence base for future policy development in this area.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