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
Shuffle algebras and vertex models. Shuffle algebras are important new mathematical structures that offer a new approaches and techniques to solve outstanding open problems in a variety of branches of mathematics, including mathematical physics, algebraic geometry and combinatorics. This project proposes to find solutions to key open problems using connections between shuffle algebras and integrable lattice models. The expected outcomes include (i) a new framework of shuffle algebra techniques t ....Shuffle algebras and vertex models. Shuffle algebras are important new mathematical structures that offer a new approaches and techniques to solve outstanding open problems in a variety of branches of mathematics, including mathematical physics, algebraic geometry and combinatorics. This project proposes to find solutions to key open problems using connections between shuffle algebras and integrable lattice models. The expected outcomes include (i) a new framework of shuffle algebra techniques to solve challenging research problems in mathematical physics and statistical mechanics, (ii) practical and computationally feasible constructions of shuffle algebras using vertex models, (iii) solutions to unresolved spectral problems of open quantum systems.Read moreRead less
Matrix product multi-variable polynomials from quantum algebras. This project aims to expand the theory of polynomials and develop generalised polynomial families using connections to affine and toroidal algebras. Many combinatorial and computational problems in pure and applied mathematics as well as mathematical physics can be solved using polynomials in many variables, such as Macdonald polynomials. This project is anticipated to address the current difficulty of implementing symmetric and no ....Matrix product multi-variable polynomials from quantum algebras. This project aims to expand the theory of polynomials and develop generalised polynomial families using connections to affine and toroidal algebras. Many combinatorial and computational problems in pure and applied mathematics as well as mathematical physics can be solved using polynomials in many variables, such as Macdonald polynomials. This project is anticipated to address the current difficulty of implementing symmetric and non-symmetric polynomials in symbolic algebra packages by developing completely new algorithms. New understanding from the project is expected to facilitate challenging computational problems of measurable quantities in quantum systems.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE210101264
Funder
Australian Research Council
Funding Amount
$342,346.00
Summary
Toroidal quantum groups, integrable models and applications. Modelling systems of quantum and classical mechanics usually relies on computationally expensive numerical methods. Such methods typically provide raw answers and give little insight. In contrast, a special class of modelling based on quantum integrability provides us with a variety of analytic tools thanks to connections with algebra, geometry and combinatorics. The project aims to study quantum integrability with the help of new exci ....Toroidal quantum groups, integrable models and applications. Modelling systems of quantum and classical mechanics usually relies on computationally expensive numerical methods. Such methods typically provide raw answers and give little insight. In contrast, a special class of modelling based on quantum integrability provides us with a variety of analytic tools thanks to connections with algebra, geometry and combinatorics. The project aims to study quantum integrability with the help of new exciting developments in toroidal quantum groups. The anticipated outcomes include constructions of new models, developing analytic methods and computer algebra packages. These results are expected to facilitate challenging computational problems in modelling of quantum and classical systems.Read moreRead less
Expanding and linking random matrix theory. Fundamental to random matrix theory are certain universality laws, holding in scaling limits to infinite matrix size. A basic question is to quantify the rate of convergence to the universal laws. The analysis of data for the Riemann zeros from prime number theory, and of the spectral form factor probe of chaos in black hole physics, are immediate applications. An analysis involving integrable structures holding for finite matrix size and their asympt ....Expanding and linking random matrix theory. Fundamental to random matrix theory are certain universality laws, holding in scaling limits to infinite matrix size. A basic question is to quantify the rate of convergence to the universal laws. The analysis of data for the Riemann zeros from prime number theory, and of the spectral form factor probe of chaos in black hole physics, are immediate applications. An analysis involving integrable structures holding for finite matrix size and their asymptotics is proposed, allowing the rate to be quantified for a large class of model
ensembles, and providing predictions in the various applied settings. The broad project is to be networked with researchers in the Asia-Oceania region, with the aim of establishing leadership status for Australia.Read moreRead less
Transformative simulation techniques for complex polymer networks. The study of long chain polymers like DNA using computer simulations has uncovered exciting insights over many years. Generally these have been limited to simple topologies, interactions, and environments. This project aims to develop the next generation of simulation techniques to tackle a new frontier of polymer models, including those with complex topologies like stars, knots, and links, which have hitherto been inaccessible. ....Transformative simulation techniques for complex polymer networks. The study of long chain polymers like DNA using computer simulations has uncovered exciting insights over many years. Generally these have been limited to simple topologies, interactions, and environments. This project aims to develop the next generation of simulation techniques to tackle a new frontier of polymer models, including those with complex topologies like stars, knots, and links, which have hitherto been inaccessible. Expected outcomes include new simulation methods which harness modern computational clusters, leading to greater understanding of polymers with complex topologies and in complicated environments. Important elements of biological processes may be discovered, such as how polymer structure affects DNA transcription.Read moreRead less
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
Unifying discrete and continuous methods in quantum information theory. This project aims to address a critical gap in quantum information theory by unifying the way that both discrete and continuous quantum systems are represented in mathematical models. This project expects to generate new knowledge in quantum information science by using cutting-edge mathematical tools and insights from signal processing theory. Expected outcomes of this project include a new mathematical framework for use in ....Unifying discrete and continuous methods in quantum information theory. This project aims to address a critical gap in quantum information theory by unifying the way that both discrete and continuous quantum systems are represented in mathematical models. This project expects to generate new knowledge in quantum information science by using cutting-edge mathematical tools and insights from signal processing theory. Expected outcomes of this project include a new mathematical framework for use in quantum science and technology development. This should provide significant benefits, such as new ways to efficiently simulate certain quantum processes on ordinary computers and novel approaches to handling noise in quantum computers.Read moreRead less
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
Visualisation of multidimensional physics data. This project aims to link multi-parameter models used in physics to explore experimental data, and statistical tools for multivariate analysis and visualisation. It addresses an important gap in the understanding of phenomenological physics analyses containing many simultaneously important parameters. This is expected to improve the understanding of results in multi-parameter models.