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
Discovery Early Career Researcher Award - Grant ID: DE210101323
Funder
Australian Research Council
Funding Amount
$345,448.00
Summary
Random tensors and random matrices: interactions and applications. This project aims at improving knowledge on probabilistic objects having applications in, for instance, mathematical-physics, statistical physics, quantum gravity and data science. In doing so, we expect to produce new mathematical results by building upon both classical approaches and innovative ones. In particular, on one hand, the extension of classical graphical methods will be developed and, on another hand, generalized prob ....Random tensors and random matrices: interactions and applications. This project aims at improving knowledge on probabilistic objects having applications in, for instance, mathematical-physics, statistical physics, quantum gravity and data science. In doing so, we expect to produce new mathematical results by building upon both classical approaches and innovative ones. In particular, on one hand, the extension of classical graphical methods will be developed and, on another hand, generalized probability theories will be used to provide new insights. The expected outcomes include a better understanding of the generic properties of quantum states. This should significantly benefit to mathematicians and physicists whose models use those objects and may impact the broader community of engineers and technicians.Read moreRead less
From quantum integrable systems to algebraic geometry and combinatorics. The purpose of this project is to investigate the deep connections that have recently emerged between the study of an area of mathematical physics (quantum integrable systems) and subjects of pure mathematics (enumerative and algebraic combinatorics, and algebraic geometry). These connections have a common root, which this project plans to reveal using novel methods coming from quantum integrability. This approach is expect ....From quantum integrable systems to algebraic geometry and combinatorics. The purpose of this project is to investigate the deep connections that have recently emerged between the study of an area of mathematical physics (quantum integrable systems) and subjects of pure mathematics (enumerative and algebraic combinatorics, and algebraic geometry). These connections have a common root, which this project plans to reveal using novel methods coming from quantum integrability. This approach is expected to illuminate these subjects leading to a new unified and interdisciplinary picture, and to resolve important open problems in the study of certain algebraic varieties and of their cohomology in the theory of symmetric functions, and related counting problems.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE190101231
Funder
Australian Research Council
Funding Amount
$390,000.00
Summary
Integrable systems from gauge theories. This project aims to construct and describe three important integrable systems and to produce new significant results in representation theory and mathematical physics. The focus of the project is the theory of integrable systems, a breakthrough area in modern mathematics, and it will use tools from geometric representation theory to study the interrelation between geometry and mathematical physics. Originating from string theory, the project will not only ....Integrable systems from gauge theories. This project aims to construct and describe three important integrable systems and to produce new significant results in representation theory and mathematical physics. The focus of the project is the theory of integrable systems, a breakthrough area in modern mathematics, and it will use tools from geometric representation theory to study the interrelation between geometry and mathematical physics. Originating from string theory, the project will not only provide a deeper understanding of the universe, it will also train young mathematicians and other highly qualified individuals with the potential to make a significant impact to technology, security, and the economy though their specialised skills.Read moreRead less
Tantalizer algebras and generalized lattice models. This project exploits underlying symmetry to characterise components and flow patterns in network configurations. The project will develop tools for analysis and optimisation of systems of interacting nodes which can arise in materials, computing networks, and any social or industrial contexts with communication or product transfer between nodes.
Finite dimensional integrable systems and differential geometry. Mathematical models of many processes in science (physics, engineering) and in the real world (nature, economics) are governed by complicated systems of differential equations. An important, distinguished class of such models is described by integrable systems, the systems for which one can provide a comprehensive qualitative picture, and in many cases, a complete solution. Using recently developed, powerful methods of integrable s ....Finite dimensional integrable systems and differential geometry. Mathematical models of many processes in science (physics, engineering) and in the real world (nature, economics) are governed by complicated systems of differential equations. An important, distinguished class of such models is described by integrable systems, the systems for which one can provide a comprehensive qualitative picture, and in many cases, a complete solution. Using recently developed, powerful methods of integrable systems and differential geometry, this project will focus on a range of important, interconnected theoretical problems in both disciplines. The expected outcomes will provide new, deep, mathematically and physically significant results which will lead to applications and developments across a range of fields.Read moreRead less
Quantum Spectra. Fundamental quantum processes will play a key role in emerging technologies in the twenty-first century across diverse industries including quantum information technology, quantum computers and electronics, quantum optics, nanoscale quantum microscopes and superconductor technology. Australia has a strong base of expertise in the underpinning quantum disciplines. This project in strategic basic research within mathematical physics will develop a comprehensive and consistent math ....Quantum Spectra. Fundamental quantum processes will play a key role in emerging technologies in the twenty-first century across diverse industries including quantum information technology, quantum computers and electronics, quantum optics, nanoscale quantum microscopes and superconductor technology. Australia has a strong base of expertise in the underpinning quantum disciplines. This project in strategic basic research within mathematical physics will develop a comprehensive and consistent mathematical description of quantum processes. This research will lead to a deeper understanding of quantum processes, keep Australia at the leading edge of international developments and increase Australia's capacity to develop and implement these new technologies.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE120101375
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
The Tutte polynomial of a graph: correlations, approximations and applications. The Tutte polynomial is a mathematical function of central importance to diverse fields of research, such as network reliability and statistical mechanics, that involve natural (and often difficult) counting problems. This project aims to obtain useful close approximations of this function with immediate applications in all these research fields.
Energy, Cosmic Censorship and Black Hole Stability. Human progress is achieved by confronting fundamental questions, at the leading edge of knowledge. This project will lead to better understanding of space-time physics, and of the properties of singular solutions of non-linear hyperbolic equations. Such equations govern a wide range of physical phenomena, including fluid flow, weather and electromagnetic fields.
Discovery Early Career Researcher Award - Grant ID: DE160100250
Funder
Australian Research Council
Funding Amount
$299,436.00
Summary
Advanced methods in combinatorial geometry. This project aims to harness new techniques to solve some challenging open problems related to visibility among sets of points. Combinatorial geometry is the mathematical study of the structure of arrangements of points, lines and other geometric objects in space. Many modern technologies require computation with such geometric data, from computer graphics to robotics and computer vision. Advances in the computational techniques that these technologies ....Advanced methods in combinatorial geometry. This project aims to harness new techniques to solve some challenging open problems related to visibility among sets of points. Combinatorial geometry is the mathematical study of the structure of arrangements of points, lines and other geometric objects in space. Many modern technologies require computation with such geometric data, from computer graphics to robotics and computer vision. Advances in the computational techniques that these technologies use are underpinned by mathematical theory. The last five years has seen major breakthroughs in combinatorial geometry, along with the development of ground-breaking new techniques. Solutions to current problems using these techniques are likely to lead to further theoretical advances and insights.Read moreRead less