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.
Advanced numerical and analytical techniques for exact studies in combinatorics and statistical mechanics. Exactly solved models are of immense importance in all areas of the theoretical sciences and play important roles in our understanding of complex natural and social phenomena. This project aims to develop powerful new methods that will enable mathematicians and physicists to greatly expand the types of models for which we can find a solution.
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.
Characteristic polynomials in random matrix theory. Random matrix theory is the subject of an active international research effort, due to its broad range of applications including the statistical analysis of high-dimensional data sets, wireless communication, and the celebrated Riemann zeros in prime number theory. Characteristic polynomials will be used to focus an attack on these problems.
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.
The connection between discrete holomorphicity and Yang-Baxter integrability. This project will develop and apply the mathematical theory underlying the rigorous study of phase transitions and critical phenomena, which defines what we know about 'everyday' matter and its transformations. The project will also contribute to training in an area for which Australia has an outstanding international reputation.
A synthesis of random matrix theory for applications in mathematics, physics and engineering. Random matrix theory, matrix theory where the elements are random, or the matrix chosen from an ensemble, is driven by its ever expanding range of applications, and the richness of the mathematics being uncovered. These applications include topics of acknowledged modern day importance, for example quantum information theory, wireless communication, data analysis, signal processing and the study of algor ....A synthesis of random matrix theory for applications in mathematics, physics and engineering. Random matrix theory, matrix theory where the elements are random, or the matrix chosen from an ensemble, is driven by its ever expanding range of applications, and the richness of the mathematics being uncovered. These applications include topics of acknowledged modern day importance, for example quantum information theory, wireless communication, data analysis, signal processing and the study of algorithms. Buoyed by promising preliminary investigations, this project aims to draw together seemingly disparate techniques to tackle problems from such topics. In addition to providing solutions to these problems, these methods are expected to provide inspiration for fellow researchers.Read moreRead less
Design, analysis and application of Monte Carlo algorithms in statistical mechanics. Monte Carlo methods provide a powerful computational tool with an enormous range of applications. However when applied in statistical mechanics they typically suffer from severe critical slowing-down, so that their computational efficiency tends rapidly to zero as a critical point is approached. We will develop novel, more efficient Monte Carlo algorithms, to simulate a range of models in statistical mechanics a ....Design, analysis and application of Monte Carlo algorithms in statistical mechanics. Monte Carlo methods provide a powerful computational tool with an enormous range of applications. However when applied in statistical mechanics they typically suffer from severe critical slowing-down, so that their computational efficiency tends rapidly to zero as a critical point is approached. We will develop novel, more efficient Monte Carlo algorithms, to simulate a range of models in statistical mechanics and back this up with rigorous mathematical analysis proving that their results can be trusted.Read moreRead less
Critical behaviour of lattice models of spin systems. A common feature of complex systems is that simple, short-range forces can produce long-range effects. Lattice spin systems are paradigms of this phenomenon, and this project proposes to solve a number of outstanding and significant problems in these systems. This will enhance our understanding of complex systems in general, and key models in particular.
Interplay of Topology and Geometry in Polymeric Critical Phenomena. This project aims to develop new understanding of key topologically driven behaviour in complex polymers such as DNA. The mathematical modelling of the rich geometric behaviour of long-chain polymers and how they change their shape in response to the environment has provided a framework for our understanding of these ubiquitous molecules. Complex polymers such as ring polymers and DNA display topological properties such as knott ....Interplay of Topology and Geometry in Polymeric Critical Phenomena. This project aims to develop new understanding of key topologically driven behaviour in complex polymers such as DNA. The mathematical modelling of the rich geometric behaviour of long-chain polymers and how they change their shape in response to the environment has provided a framework for our understanding of these ubiquitous molecules. Complex polymers such as ring polymers and DNA display topological properties such as knotting and linking. Recent experiments twisting DNA demonstrate novel phenomena precipitated by the interplay of topological and geometric properties. Using advanced mathematical and computational techniques, the project aims to explain how topological constraints and changes disturb key polymer behaviour.Read moreRead less