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
Operator Integrals and Derivatives. The project is a contribution to the study of non-commutative differential and integral calculus. The novelty of the present project lies in the study of smoothness properties of functions whose domains and ranges are spaces of unbounded, non-commuting operators on some Hilbert space. Our general approach will be based on a detailed investigation of properties of double operator integrals, which permit smoothness estimates of operator-functions. It can be expe ....Operator Integrals and Derivatives. The project is a contribution to the study of non-commutative differential and integral calculus. The novelty of the present project lies in the study of smoothness properties of functions whose domains and ranges are spaces of unbounded, non-commuting operators on some Hilbert space. Our general approach will be based on a detailed investigation of properties of double operator integrals, which permit smoothness estimates of operator-functions. It can be expected that the new techniques generated will find further application in areas of mathematical physics and non-commutative geometry related to quantized calculus.
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Non-commutative analysis and differential calculus. This project is in an area of central mathematical importance and will lead to important scientific advances that will keep Australia at the forefront internationally in this field of research. There is an emphasis on international networking and we will collaborate with leading researchers in USA and France.
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
Discovery Early Career Researcher Award - Grant ID: DE160100525
Funder
Australian Research Council
Funding Amount
$392,053.00
Summary
Index Theory for Spaces with Symmetries. This project aims to study spaces with symmetries, which are important geometric models in topology and representation theory. The project plans to conduct research into geometric approaches to the representation theory of groups using KK theory and index theory from the perspective of operator algebra. The expected outcomes of this project are constructions of new topological invariants and their geometric formulas for spaces with symmetries and applicat ....Index Theory for Spaces with Symmetries. This project aims to study spaces with symmetries, which are important geometric models in topology and representation theory. The project plans to conduct research into geometric approaches to the representation theory of groups using KK theory and index theory from the perspective of operator algebra. The expected outcomes of this project are constructions of new topological invariants and their geometric formulas for spaces with symmetries and applications in representation theory.Read moreRead less
Novel geometric invariants. Quantum theory is the language of fundamental physics, it describes the small scale structure of matter and possibly space-time. Sophisticated models in condensed matter physics and string theory have exposed geometric and topological structure as basic building blocks of the theory. Issues thrown up by quantum theory are very similar to, and have provided techniques to solve, problems in the geometry of three and four dimensional manifolds. Exciting two way exchanges ....Novel geometric invariants. Quantum theory is the language of fundamental physics, it describes the small scale structure of matter and possibly space-time. Sophisticated models in condensed matter physics and string theory have exposed geometric and topological structure as basic building blocks of the theory. Issues thrown up by quantum theory are very similar to, and have provided techniques to solve, problems in the geometry of three and four dimensional manifolds. Exciting two way exchanges of methods, problems and solutions have emerged. This project aims to settle fundamental questions in the interaction between these two fields.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE120100901
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Deformation quantisation and index theory for semi-simple groups. Deformation quantisation is a mathematical technique for describing the counter-intuitive geometry of quantum physics as a small variation of classical geometry as Newton would have known it. This project will apply the same techniques to solve fundamental mathematical problems in the study of symmetries.
Special Research Initiatives - Grant ID: SR0354466
Funder
Australian Research Council
Funding Amount
$20,000.00
Summary
Mathematics in Contemporary Science. The Mathematics in Contemporary Science Research Network brings contemporary methods of non-linear analysis and differential equations, geometric reasoning and relevant algebraic and topological ideas to enrich six application areas in modern science: Complex Systems, Computer Vision, Optimal Transportation, Nanotechnology, Physics and Shortest Networks. MiCS will develop both the mathematics and the application areas in parallel. It will focus on postgradu ....Mathematics in Contemporary Science. The Mathematics in Contemporary Science Research Network brings contemporary methods of non-linear analysis and differential equations, geometric reasoning and relevant algebraic and topological ideas to enrich six application areas in modern science: Complex Systems, Computer Vision, Optimal Transportation, Nanotechnology, Physics and Shortest Networks. MiCS will develop both the mathematics and the application areas in parallel. It will focus on postgraduate training through workshops, summer schools and web based resources and build long-term international collaborations with EU networks and NSERC, NSF and EPSRC institutes as well as bringing together academic and industry leaders.Read moreRead less
New approaches to index theory. The laws of nature are often expressed by differential equations, involving their rates of change. If 'elliptic,' they have an 'index,' which is the number of solutions minus the number of constraints imposed. The Atiyah-Singer index theorem gives a striking calculation of this "index'. An extension is Fractional Index Theory, which has received international attention, having solved a fundamental open problem. A central aim is to investigate this further. I will ....New approaches to index theory. The laws of nature are often expressed by differential equations, involving their rates of change. If 'elliptic,' they have an 'index,' which is the number of solutions minus the number of constraints imposed. The Atiyah-Singer index theorem gives a striking calculation of this "index'. An extension is Fractional Index Theory, which has received international attention, having solved a fundamental open problem. A central aim is to investigate this further. I will assist beginners to navigate to the cutting edge of research through workshops, spring-schools and supervision. Benefits include the enhancement of Australia's position in the forefront of international research.Read moreRead less
Multiparameter Harmonic Analysis: Weighted Estimates for Singular Integrals. This project aims to study advanced harmonic analysis concerning multiparameter theory and related topics. Harmonic analysis lies at the intersection of the frontiers of many branches of mathematics. It is fundamental to the study of operator theory and partial differential equations which has wide applications in many fields such as mathematical modelling, probability and number theory. This project aims to solve a num ....Multiparameter Harmonic Analysis: Weighted Estimates for Singular Integrals. This project aims to study advanced harmonic analysis concerning multiparameter theory and related topics. Harmonic analysis lies at the intersection of the frontiers of many branches of mathematics. It is fundamental to the study of operator theory and partial differential equations which has wide applications in many fields such as mathematical modelling, probability and number theory. This project aims to solve a number of open problems at the frontier of research in modern harmonic analysis including estimates on multilinear operators with nonsmooth kernels and advanced multiparameter theory on product spaces.Read moreRead less