A Bayesian State Space Methodology for Forecasting Stock Market Volatility and Associated Time-varying Risk Premia. Accurate prediction of stock market volatility is critical for effective financial risk management. Along with information on volatility embedded in historical stock market returns, the prices of options written on the underlying stocks also reflect the option market's assessment of future volatility. This project will exploit this dual data source in a completely new way, using it ....A Bayesian State Space Methodology for Forecasting Stock Market Volatility and Associated Time-varying Risk Premia. Accurate prediction of stock market volatility is critical for effective financial risk management. Along with information on volatility embedded in historical stock market returns, the prices of options written on the underlying stocks also reflect the option market's assessment of future volatility. This project will exploit this dual data source in a completely new way, using it to produce forecasts of both volatility itself and the premia factored into asset prices as a result of traders' perceptions of volatility risk. State-of-the-art statistical methods will be used to produce up-dates of the probability of extreme volatility and/or extreme risk aversion, as new market data becomes available each trading day.Read moreRead less
A mathematical analysis of the influence of small scale inhomogeneities on the evolution of the universe. A fundamental unresolved problem in modern cosmology is to quantify the influence of small-scale inhomogeneities on the evolution of the universe. This project will develop the mathematical techniques required to resolve this question. In addition, these techniques will have important applications to the analysis of astronomical data.
Solvability and universality in stochastic processes. Exactly solvable stochastic processes are an important area of mathematical research, with cross-disciplinary links to quantum physics, quantum algebras and probability theory. These processes can be used to model a variety of real-world phenomena such as crystal growth and polymers in random media. This project aims to significantly expand our knowledge of exactly solvable stochastic processes by extending them to new algebraic frameworks. A ....Solvability and universality in stochastic processes. Exactly solvable stochastic processes are an important area of mathematical research, with cross-disciplinary links to quantum physics, quantum algebras and probability theory. These processes can be used to model a variety of real-world phenomena such as crystal growth and polymers in random media. This project aims to significantly expand our knowledge of exactly solvable stochastic processes by extending them to new algebraic frameworks. Among the outcomes of the project, we expect to identify new probabilistic structures which go beyond the famous Gaussian universality class. These theoretical developments allow better prediction of randomly growing interfaces, which encompass a range of phenomena from tumour growth to forest fires.Read moreRead less
Representation theory of diagram algebras and logarithmic conformal field theory. Generalized models of polymers and percolation are notoriously difficult to handle mathematically, but can be described and solved using diagram algebras and logarithmic conformal field theory. Potential applications include polymer-like materials, filtering of drinking water, spatial spread of epidemics and bushfires, and tertiary recovery of oil.
Design, analysis and application of Monte Carlo methods in statistical mechanics. Statistical mechanics is a general framework for studying complex systems and Monte Carlo methods are an important computational tool in such studies. This project will develop new, vastly more efficient, Monte Carlo methods for problems in statistical mechanics, and will apply these methods to real-world problems such as urban traffic flow.
Representation theory in exactly solvable systems. This project aims to develop the representation theory of Lie and generalised Lie algebras related to exactly solvable models. The project will exploit several innovative ideas on the structure of quadratic algebras, Casimir invariants, differential operator realisations, roots systems, characters and indecomposable representations. This will give fundamental mathematical insight and allow the construction of new, exactly solvable models. This w ....Representation theory in exactly solvable systems. This project aims to develop the representation theory of Lie and generalised Lie algebras related to exactly solvable models. The project will exploit several innovative ideas on the structure of quadratic algebras, Casimir invariants, differential operator realisations, roots systems, characters and indecomposable representations. This will give fundamental mathematical insight and allow the construction of new, exactly solvable models. This will have an impact on theoretical physics as exactly solvable models play a central role in our understanding of a plethora of physical phenomena.Read moreRead less
Computational studies of soft matter. Soft matter systems such as colloidal suspensions and polymers are ubiquitous in nature, and industrially important. For colloidal systems, specifically hard spheres, this project will utilise new algorithms to attack long standing questions about the nature of the virial series. For self-avoiding walks and related models of polymers, research studies have recently developed radically improved Monte Carlo simulation algorithms. These algorithms will enable t ....Computational studies of soft matter. Soft matter systems such as colloidal suspensions and polymers are ubiquitous in nature, and industrially important. For colloidal systems, specifically hard spheres, this project will utilise new algorithms to attack long standing questions about the nature of the virial series. For self-avoiding walks and related models of polymers, research studies have recently developed radically improved Monte Carlo simulation algorithms. These algorithms will enable this project to simulate polymers which may be as long as DNA, and to calculate physical properties with unprecedented precision. The software developed for studying polymers will be released as an open source software library which will revolutionise the field of polymer simulation.Read moreRead less
Improved theory and practice in econometric modelling of nonlinear spatial time series. Modern Australia faces many challenges in economic and global climate changes, which require advanced statistical technologies in modeling and forecasting of econometric spatial time series data. This project will provide flexible models and methods that enable practitioners to more accurately measure and manage economic and climatic risks.
Logarithmic conformal field theory and the 4D/2D correspondence. Conformal field theory provides powerful methods for attacking problems in theoretical physics and furnishes beautiful connections between seemingly disparate branches of pure mathematics. This proposal aims to greatly expand our knowledge of the logarithmic conformal field theories that have recently witnessed a resurgence of interest in physics. Advancing these theories is crucial to progress in high-energy physics and pure mathe ....Logarithmic conformal field theory and the 4D/2D correspondence. Conformal field theory provides powerful methods for attacking problems in theoretical physics and furnishes beautiful connections between seemingly disparate branches of pure mathematics. This proposal aims to greatly expand our knowledge of the logarithmic conformal field theories that have recently witnessed a resurgence of interest in physics. Advancing these theories is crucial to progress in high-energy physics and pure mathematics. Expected outcomes include a completely new understanding of the mathematical structure of these theories which will, in turn, facilitate applications in 4D gauge theory. This will boost research capacity and further cement Australia's reputation as an international leader in mathematical physics research.Read moreRead less
Quasi-subtractive varieties: a unified framework for substructural, modal and quantum logic. An algebraic theory is proposed that provides a common umbrella for a plethora of non-classical logics. At the same time, it identifies a core that these logics share with classical algebras.