New methods in spectral geometry. This project aims to use methods from mathematical scattering theory to resolve problems in the spectral analysis and index theory of differential operators. Both areas underpin the theoretical understanding of physical materials at micro length scales where quantum phenomena dominate. The project will develop new mathematical results in spectral analysis and geometry, and apply its results to theoretical models of quantum phenomena whose spectral properties are ....New methods in spectral geometry. This project aims to use methods from mathematical scattering theory to resolve problems in the spectral analysis and index theory of differential operators. Both areas underpin the theoretical understanding of physical materials at micro length scales where quantum phenomena dominate. The project will develop new mathematical results in spectral analysis and geometry, and apply its results to theoretical models of quantum phenomena whose spectral properties are at the limit of the range of mathematical techniques. Solving these problems is expected to influence non-commutative analysis.Read moreRead less
Quantum symmetries: mathematical models for topological matter. This project aims to investigate quantum symmetries, new mathematical objects which allow an algebraic description of topological phases of matter. The project expects to bridge the current gap between our mathematical and physical understandings of these topological phases of matter. The project will develop innovative tools for analysing and constructing new exotic symmetries, and provide an extensive survey of examples. It is exp ....Quantum symmetries: mathematical models for topological matter. This project aims to investigate quantum symmetries, new mathematical objects which allow an algebraic description of topological phases of matter. The project expects to bridge the current gap between our mathematical and physical understandings of these topological phases of matter. The project will develop innovative tools for analysing and constructing new exotic symmetries, and provide an extensive survey of examples. It is expected to build national research capacity in an emerging field and put Australia at the forefront of the mathematics of topological matter.Read moreRead less
Coarse Geometry: a novel approach to the Callias index & topological matter. Coarse geometry is the study of the large-scale structure of metric spaces, in terms of operator algebras. This project aims to use coarse geometry to develop novel approaches to Callias index theory and its applications, and to topological phases of matter, where the Nobel Prize in physics in 2016 was awarded. This will yield new techniques in index theory and other areas, and solutions to several important problems. O ....Coarse Geometry: a novel approach to the Callias index & topological matter. Coarse geometry is the study of the large-scale structure of metric spaces, in terms of operator algebras. This project aims to use coarse geometry to develop novel approaches to Callias index theory and its applications, and to topological phases of matter, where the Nobel Prize in physics in 2016 was awarded. This will yield new techniques in index theory and other areas, and solutions to several important problems. Outcomes include a noncompact generalisation of the famous Guillemin-Sternberg conjecture that quantisation commutes with reduction, and new models of topological phases of matter in terms of K-theory of operator algebras. This project will benefit Australia by reinforcing its position in these highly active areas in science.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.
Asymptotic Geometric Analysis and Learning Theory. Learning Theory is used in various real-world applications in diverse research areas, ranging from Biology (e.g. DNA sequencing) to Information Sciences. Therefore, having a deep understanding of fundamental questions in Learning Theory, and in particular, pin-pointing the parameters that make a learning problem hard would have a significant practical impact. This projects aims to achieve this goal, and in addition, we expect it would have a hig ....Asymptotic Geometric Analysis and Learning Theory. Learning Theory is used in various real-world applications in diverse research areas, ranging from Biology (e.g. DNA sequencing) to Information Sciences. Therefore, having a deep understanding of fundamental questions in Learning Theory, and in particular, pin-pointing the parameters that make a learning problem hard would have a significant practical impact. This projects aims to achieve this goal, and in addition, we expect it would have a high theoretical value, as the questions we shall address are of independent interest to pure mathematicians.
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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
Topological stability from spectral analysis. The aim is to use mathematical scattering theory to find and study new topological features of the spectra of linear transformations on Hilbert space. The significance derives from mathematical models of low temperature conducting quantum materials. These have revealed `topological phases of matter' that are stable with respect to a range of variations in the parameters that determine the system. The stability is desired for applications to quantum ....Topological stability from spectral analysis. The aim is to use mathematical scattering theory to find and study new topological features of the spectra of linear transformations on Hilbert space. The significance derives from mathematical models of low temperature conducting quantum materials. These have revealed `topological phases of matter' that are stable with respect to a range of variations in the parameters that determine the system. The stability is desired for applications to quantum devices. Our results will give topological stability from the scattering spectrum, a feature not previously seen. The benefits stem from new results in mathematical scattering theory with a primary novelty being the analysis of ``zero energy resonances'' in mathematical models of graphene.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE150101647
Funder
Australian Research Council
Funding Amount
$330,000.00
Summary
Symplectic solvmanifolds and their friends. Symplectic geometry is the mathematical foundation of classical mechanics and quantum theory. The symmetry group of a physical system determines the conservation laws governing its behaviour. This project aims to advance the understanding of a large class of these symmetry groups and their associated symplectic geometries, which are called symplectic solvmanifolds. The project aims to: determine the topological properties of symplectic solvmanifolds as ....Symplectic solvmanifolds and their friends. Symplectic geometry is the mathematical foundation of classical mechanics and quantum theory. The symmetry group of a physical system determines the conservation laws governing its behaviour. This project aims to advance the understanding of a large class of these symmetry groups and their associated symplectic geometries, which are called symplectic solvmanifolds. The project aims to: determine the topological properties of symplectic solvmanifolds as encoded in their fundamental groups; their geometric properties in the form of holonomy groups; and the algebraic properties of their symplectic algebras. The project endeavours to classify the building blocks of symplectic geometry.Read moreRead less
Special Research Initiatives - Grant ID: SR0354716
Funder
Australian Research Council
Funding Amount
$10,000.00
Summary
Energetically Open Systems Research Network Study. Conceptual frameworks arising in the physical sciences, such as non-equilibrium statistical mechanics and thermodynamics, synergetics, chaos and dynamical systems theory, are seminal in the emerging science of complexity. This study will lay the groundwork for a network to link Australian and overseas research on these fundamental concepts, and their application within the context of entropy-producing systems vital to the long-term sustainabilit ....Energetically Open Systems Research Network Study. Conceptual frameworks arising in the physical sciences, such as non-equilibrium statistical mechanics and thermodynamics, synergetics, chaos and dynamical systems theory, are seminal in the emerging science of complexity. This study will lay the groundwork for a network to link Australian and overseas research on these fundamental concepts, and their application within the context of entropy-producing systems vital to the long-term sustainability of the earth - oceans, atmosphere, biosphere, CO2-free energy production, space and solar environment. The network would facilitate the development of young investigators and be linked into wider complex systems networks such as the CSIRO Centre for Complex Systems Science.Read moreRead less
New Geometric and Entropy Techniques for Differential Equations. The three main practical outcomes of this mathematical research will be better predictability of salt movement responsible for land degradation, better predictability of surface evolution of microelectronic components in nanoscale technology and an open source computer package that harnesses new and powerful geometrical techniques to solve differential equations. The project will train the next generation of researchers in the math ....New Geometric and Entropy Techniques for Differential Equations. The three main practical outcomes of this mathematical research will be better predictability of salt movement responsible for land degradation, better predictability of surface evolution of microelectronic components in nanoscale technology and an open source computer package that harnesses new and powerful geometrical techniques to solve differential equations. The project will train the next generation of researchers in the mathematical modelling of critical physical processes and it will bring international experts to Australia to work on these vital problems.Read moreRead less