Multivariate polynomials: combinatorics and applications. Many combinatorial and computational problems in pure and applied mathematics as well as mathematical physics can be solved using polynomials in many variables. This proposal will expand our understanding of the poorly understood class of non-symmetric polynomials by studying their novel combinatorial structure. The outcomes will address the current difficulty of implementing non-symmetric polynomials in symbolic algebra packages by devel ....Multivariate polynomials: combinatorics and applications. Many combinatorial and computational problems in pure and applied mathematics as well as mathematical physics can be solved using polynomials in many variables. This proposal will expand our understanding of the poorly understood class of non-symmetric polynomials by studying their novel combinatorial structure. The outcomes will address the current difficulty of implementing non-symmetric polynomials in symbolic algebra packages by developing completely new computational algorithms. Secondly, this new understanding will be used to solve several challenging mathematical enumeration problems.Read moreRead less
Mathematical models for disordered critical point theories. This project sets up a team to develop innovative techniques for fundamental advances in critical behaviour of disordered systems including the Nobel Prize winning integer quantum Hall effect. It will yield new mathematical models for disordered critical point theories, essential for the theoretical analysis of associated emerging technologies.
Mathematical structure of the quantum Rabi model. This project aims to find the mathematical structure behind the quantum Rabi model, the simplest model describing the interaction between quantum light and matter. The Rabi model is the connecting link in the essential interplay between mathematics, physics, and technological applications. Solving the mathematical structure behind it is expected to form the basis for solving related and equally important models. Such models describe a qubit, the ....Mathematical structure of the quantum Rabi model. This project aims to find the mathematical structure behind the quantum Rabi model, the simplest model describing the interaction between quantum light and matter. The Rabi model is the connecting link in the essential interplay between mathematics, physics, and technological applications. Solving the mathematical structure behind it is expected to form the basis for solving related and equally important models. Such models describe a qubit, the building block of quantum information technologies, and so could realise quantum algorithms and quantum computations.Read moreRead less
Classical and affine W-algebras. The project aims to address major mathematical problems on representations of the families of quantum groups and vertex algebras associated with Lie algebras. It aims to create new connections between representation theory and mathematical physics. The theory of quantum groups originated from solvable lattice models in statistical mechanics and has turned out to have important connections with and applications to a wide range of subjects in mathematics and physic ....Classical and affine W-algebras. The project aims to address major mathematical problems on representations of the families of quantum groups and vertex algebras associated with Lie algebras. It aims to create new connections between representation theory and mathematical physics. The theory of quantum groups originated from solvable lattice models in statistical mechanics and has turned out to have important connections with and applications to a wide range of subjects in mathematics and physics. The project aims to rely on these connections to extend and develop explicit theory of both the classical and quantum versions of the vertex algebras which are of great importance to conformal field theory and soliton spin chain models.Read moreRead less
Algebraic and computational approaches for classical and quantum systems. This project aims to use a combination of algebraic, analytic and numerical techniques to develop computational algorithms to address a range of notoriously challenging problems in the mathematical sciences. These problems involve predicting the large-scale behaviour of strongly interacting classical and quantum spin systems originating in condensed matter physics, including models of relevance to proposals for topological ....Algebraic and computational approaches for classical and quantum systems. This project aims to use a combination of algebraic, analytic and numerical techniques to develop computational algorithms to address a range of notoriously challenging problems in the mathematical sciences. These problems involve predicting the large-scale behaviour of strongly interacting classical and quantum spin systems originating in condensed matter physics, including models of relevance to proposals for topological quantum computation and the latest progress using field theory. The project outcomes will involve advances in understanding these systems from new exact results and high precision numerical estimates.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE130101067
Funder
Australian Research Council
Funding Amount
$302,540.00
Summary
New constructions of superintegrable systems and the connection with Painlevé transcendents. The research of this project will lead to deep discoveries in the field of superintegrability and expand our knowledge of their related algebraic structures, supersymmetric quantum mechanics and Painlevé transcendents. The project will generate new techniques that will be utilised in future applications of mathematical and theoretical physics.
Unified theory of Richardson-Gaudin integrability. Richardson-Gaudin systems form a class of mathematical models of interacting particles that serve as a foundation to understand important phenomena in modern physics. Being integrable, these quantum systems enable deep insights. They are tractable so as to allow for exact analysis, while being elaborate enough to exhibit complex physical properties, notably phase transitions. The international team of researchers aims to merge various approaches ....Unified theory of Richardson-Gaudin integrability. Richardson-Gaudin systems form a class of mathematical models of interacting particles that serve as a foundation to understand important phenomena in modern physics. Being integrable, these quantum systems enable deep insights. They are tractable so as to allow for exact analysis, while being elaborate enough to exhibit complex physical properties, notably phase transitions. The international team of researchers aims to merge various approaches for analysing the integrability of such models. Successful outcomes are expected to produce inventive mathematical techniques, linking a diverse range of fields of current activity and growth. The resulting unified theory is expected to open the door to exciting and innovative pathways 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
Indecomposable representation theory. The project aims to develop a systematic approach to the study and application of indecomposable representations in pure mathematics and mathematical physics. Indecomposability is a central concept in representation theory and is thus fundamental to a wide range of applications in science. Examples of important contexts considered are diagram algebras and finite and infinite-dimensional Lie algebras including the Virasoro algebra underlying conformal field t ....Indecomposable representation theory. The project aims to develop a systematic approach to the study and application of indecomposable representations in pure mathematics and mathematical physics. Indecomposability is a central concept in representation theory and is thus fundamental to a wide range of applications in science. Examples of important contexts considered are diagram algebras and finite and infinite-dimensional Lie algebras including the Virasoro algebra underlying conformal field theory. Linear algebra is a ubiquitous mathematical tool playing a pivotal role in representation theory, and the project aims to resolve outstanding fundamental issues concerning families of so-called non-diagonalisable matrices.Read moreRead less
Vertex algebras and representations of quantum groups. The project will tackle mathematical problems involving algebraic structures that have fascinated scientists for several decades, and which are of fundamental importance to theoretical physics. The research will attract talented PhD students and visiting researchers, and will enhance Australia's scientific reputation.