Harmonic analysis in rough contexts. Harmonic analysis is a set of mathematical techniques aimed at decomposing complex signals into simple pieces in a way that is reminiscent of the decomposition of sounds into harmonics. It is highly efficient in analysing signals in homogeneous media such as wave propagation through the air that underpins wireless communication technology. However, wave propagation through inhomogeneous media, such as the human body in medical imaging or the Earth in geophysi ....Harmonic analysis in rough contexts. Harmonic analysis is a set of mathematical techniques aimed at decomposing complex signals into simple pieces in a way that is reminiscent of the decomposition of sounds into harmonics. It is highly efficient in analysing signals in homogeneous media such as wave propagation through the air that underpins wireless communication technology. However, wave propagation through inhomogeneous media, such as the human body in medical imaging or the Earth in geophysical imaging, is much harder to model. Phenomena with random components, as considered in finance for instance, are also problematic. This project is an important part of an intense international research effort to develop harmonic analysis in such rough contexts.Read moreRead less
The Spectral Theory and Harmonic Analysis of Geometric Differential Operators. The project will involve mathematical research of the highest international standard in two very active and far-reaching field of mathematics: quantum chaos, and harmonic analysis. Progress in these fields will have implications in areas such as communications technology (e.g. image compression), quantum theory, and mathematical analysis (e.g. partial differential equations).
Operator algebras as models for dynamics and geometry. Operator algebra is the mathematical theory which describes quantum physics and predicts how quantum systems will behave. Through this project, the researcher's recent discoveries in operator algebra will give us new insight into the dynamics and geometry - that is, the behaviour and shape - of the quantum world.
Regularisation methods of inverse problems: theory and computation. This project aims to investigate regularisation methods for inverse problems which are ill-posed in the sense that their solutions depend discontinuously on the data. When only noisy data is available, regularisation methods define stable approximate solutions by replacing the original inverse problem with a family of well-posed neighbouring problems monitored by a so-called regularisation parameter. The project expects to devel ....Regularisation methods of inverse problems: theory and computation. This project aims to investigate regularisation methods for inverse problems which are ill-posed in the sense that their solutions depend discontinuously on the data. When only noisy data is available, regularisation methods define stable approximate solutions by replacing the original inverse problem with a family of well-posed neighbouring problems monitored by a so-called regularisation parameter. The project expects to develop purely data-driven rules to choose the regularisation parameter and show how they work in theory, and in practice. It will also develop convex framework, acceleration strategies as well as preconditioning and splitting ideas to design efficient regularisation solvers.Read moreRead less
Problems in harmonic analysis: decoupling and Bourgain-Brezis inequalities. This project in mathematics aims to study two recent, promising developments in harmonic analysis, namely Fourier decoupling and Bourgain-Brezis inequalities. The former captures how waves interfere upon superposition; the latter arose initially in the study of the Ginzburg-Landau theory of superconductors. This exciting project seeks to deliver deep insights into how different frequencies interact, and aims to develop p ....Problems in harmonic analysis: decoupling and Bourgain-Brezis inequalities. This project in mathematics aims to study two recent, promising developments in harmonic analysis, namely Fourier decoupling and Bourgain-Brezis inequalities. The former captures how waves interfere upon superposition; the latter arose initially in the study of the Ginzburg-Landau theory of superconductors. This exciting project seeks to deliver deep insights into how different frequencies interact, and aims to develop powerful new tools to advance the study of partial differential equations and analytic number theory. This Future Fellowship should benefit Australia by improving our scientific capability. It will bring world-class researchers to Australia for collaboration, and put Australia at the forefront of first rate research.
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Increasing internet energy and cost efficiency by improving higher-layer protocols. Australians rely heavily on our telecommunications infrastructure due to our geographic dispersion. We are also very susceptible to climate change, given our reliance on agriculture. Information technology is consuming a rapidly increasing fraction of our power and our budget. This research will help to reverse both those trends, by finding novel and practical ways to use our infrastructure more efficiently, and ....Increasing internet energy and cost efficiency by improving higher-layer protocols. Australians rely heavily on our telecommunications infrastructure due to our geographic dispersion. We are also very susceptible to climate change, given our reliance on agriculture. Information technology is consuming a rapidly increasing fraction of our power and our budget. This research will help to reverse both those trends, by finding novel and practical ways to use our infrastructure more efficiently, and to minimise its energy use. This will enable the Australian telecommunications industry to provide better service (including to Australian industries and rural communities) at lower economic and environmental cost. This project will put Australia on the international stage as a leading contributor to energy-efficient internet technology.Read moreRead less
Curvature flows and spectral estimates. Curvature flows are a class of geometrically motivated equations, modelled on the heat equation. Recently, researchers have developed new methods for studying the regularity of solutions to these equations, and applied them to a different problem, that of estimating quantities depending on the smaller eigenvalues of a Schroedinger operator. This project builds on the early success of this research and will produce a new understanding of the behaviour of ei ....Curvature flows and spectral estimates. Curvature flows are a class of geometrically motivated equations, modelled on the heat equation. Recently, researchers have developed new methods for studying the regularity of solutions to these equations, and applied them to a different problem, that of estimating quantities depending on the smaller eigenvalues of a Schroedinger operator. This project builds on the early success of this research and will produce a new understanding of the behaviour of eigenvalues, establish sharp estimates for spectral quantities, particularly on manifolds with curvature bounds, and find optimal conditions under which non-compact solutions to curvature flows are stable.Read moreRead less
Computational methods for solving modern asset pricing models. This project aims to solve a broad range of asset pricing models. Movements in asset prices affect private investors, public sector finances, wealth distribution and business activity levels. Economists have tried to build better models of asset prices, moving away from hyper-rationality and towards realistic features including heterogeneity, habit persistence and bounded rationality. These models’ additional complexity makes them di ....Computational methods for solving modern asset pricing models. This project aims to solve a broad range of asset pricing models. Movements in asset prices affect private investors, public sector finances, wealth distribution and business activity levels. Economists have tried to build better models of asset prices, moving away from hyper-rationality and towards realistic features including heterogeneity, habit persistence and bounded rationality. These models’ additional complexity makes them difficult to solve or to apply to real world problems. The project will use modern hardware and computational tools, insights from economics literature and numerical analysis to provide a set of solution methods for such asset pricing models. This is expected to improve policy analysis and decision making under uncertainty.Read moreRead less