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.
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.
Theory and Applications of Hypergeometric Series. Techniques based on hypergeometric series lie at the heart of an exciting and rapidly developing class of mathematical methods, with applications to many areas of science and engineering, such as computer science, statistics, physics, chemistry and biology.
In the past decades Australia has been at the forefront of important developments in the field, and this proposal serves to further strengthen the country's leading reputation.
Many of th ....Theory and Applications of Hypergeometric Series. Techniques based on hypergeometric series lie at the heart of an exciting and rapidly developing class of mathematical methods, with applications to many areas of science and engineering, such as computer science, statistics, physics, chemistry and biology.
In the past decades Australia has been at the forefront of important developments in the field, and this proposal serves to further strengthen the country's leading reputation.
Many of the modern methods in the theory require expertise in mathematics as well as a high level of programming skills. This combination provides a unique training ground for higher degree students aiming at careers in financial mathematics, weather/climate forecasting and internet security.
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Universal structures in stringy extra dimensions. The project aims to study properties of extra dimensions in string theory by means of techniques from supersymmetric gauge theory. This new approach makes it possible to study areas in the landscape of stringy extra dimensions that have not been accessible before. The project expects to uncover new universal features. This will have significant impact on string theory and mathematics. Expected outcomes of this project include answers to conceptua ....Universal structures in stringy extra dimensions. The project aims to study properties of extra dimensions in string theory by means of techniques from supersymmetric gauge theory. This new approach makes it possible to study areas in the landscape of stringy extra dimensions that have not been accessible before. The project expects to uncover new universal features. This will have significant impact on string theory and mathematics. Expected outcomes of this project include answers to conceptual questions in string theory, new types of extra dimensions, and new methods to compute quantum corrections in string theory. This should provide significant benefits, such as interdisciplinary collaborations at the national and international level and a strengthening of string theory in Australia.Read moreRead less
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.
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.
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
Towards higher rank logarithmic conformal field theories. This project aims to expand our knowledge of logarithmic theories. Conformal field theory provides powerful methods for attacking problems in theoretical physics and furnishes beautiful connections between seemingly disparate branches of pure mathematics. Advancing these theories is crucial to progress in statistical mechanics, string theory and various mathematical disciplines. Expected outcomes include a detailed formalism for systemati ....Towards higher rank logarithmic conformal field theories. This project aims to expand our knowledge of logarithmic theories. Conformal field theory provides powerful methods for attacking problems in theoretical physics and furnishes beautiful connections between seemingly disparate branches of pure mathematics. Advancing these theories is crucial to progress in statistical mechanics, string theory and various mathematical disciplines. Expected outcomes include a detailed formalism for systematically and rigorously analysing a wide variety of logarithmic conformal field theories so as to facilitate applications.Read moreRead less
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