Global aspects of dualities in String Theory in the presence of background fluxes. String Theory, known to the general public as the "Theory of Everything', is currently an extremely active area of research internationally. It has not only stimulated considerable interaction between mathematical physicists and mathematicians, but also increased public interest in science through television programs and books. Unfortunately, the majority of the Australian scientific community has not yet caught ....Global aspects of dualities in String Theory in the presence of background fluxes. String Theory, known to the general public as the "Theory of Everything', is currently an extremely active area of research internationally. It has not only stimulated considerable interaction between mathematical physicists and mathematicians, but also increased public interest in science through television programs and books. Unfortunately, the majority of the Australian scientific community has not yet caught up with these developments. Our recent papers, all published in premier journals in this field, have not only received widespread international attention but have also increased the profile of String Theory amongst Australia's mathematicians and mathematical physicists. The proposed project is expected to continue this trend.Read moreRead less
Symmetry in Differential Geometry. Differential geometry is a major branch of mathematics studying shape by using calculus and differential equations. This is a fundamental research project in this area, especially concerned with the interaction between geometry, differential equations, and symmetry. The mathematical notion of symmetry was already formalised early last century and nowadays lies at the very heart of mathematics and physics. Advances in this area provide essential tools in basic s ....Symmetry in Differential Geometry. Differential geometry is a major branch of mathematics studying shape by using calculus and differential equations. This is a fundamental research project in this area, especially concerned with the interaction between geometry, differential equations, and symmetry. The mathematical notion of symmetry was already formalised early last century and nowadays lies at the very heart of mathematics and physics. Advances in this area provide essential tools in basic science and unexpected technological benefits can easily arise (for example, in medical imaging). Fundamental mathematical research is absolutely necessary if Australia is to maintain a presence on the international scientific stage.
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Higher Line Bundles in Geometry and Physics. This project seeks to develop a theory of geometric objects, `higher line bundles', which realise elements of higher dimensional cohomology groups. In particular this project will develop a theory of differential geometry for these objects, allowing one to interpret differential forms representing cohomology classes as the `curvature' of a higher line bundle. This will have applications in quantum field theory and string/brane theory.
Monopoles, instantons and metrics. This Project is pure basic research in the general area of differential geometry or the study of manifolds. Manifolds are higher dimensional analogues of surfaces such as the surface of the sphere or the surface of a doughnut. This Project studies monopoles and instantons which are solutions of partial differential equations arising in physics. These solutions and the so-called moduli spaces of all solutions have been used in the last two decades by the worlds ....Monopoles, instantons and metrics. This Project is pure basic research in the general area of differential geometry or the study of manifolds. Manifolds are higher dimensional analogues of surfaces such as the surface of the sphere or the surface of a doughnut. This Project studies monopoles and instantons which are solutions of partial differential equations arising in physics. These solutions and the so-called moduli spaces of all solutions have been used in the last two decades by the worlds leading mathematicians to revolutionize the study of three and four dimensional manifolds.Read moreRead less
Characterizing and classifying ovoids, flocks and generalized quadrangles. This project lies within the framework of the classification and characterization of fundamental structures in finite geometry. This research area is the site of much international activity, in which the proposed research team plays a central role. The aim of the project is to pursue twin goals: the classification of ovoids in three dimensional projective space, a famous long-standing problem; and the classification of ce ....Characterizing and classifying ovoids, flocks and generalized quadrangles. This project lies within the framework of the classification and characterization of fundamental structures in finite geometry. This research area is the site of much international activity, in which the proposed research team plays a central role. The aim of the project is to pursue twin goals: the classification of ovoids in three dimensional projective space, a famous long-standing problem; and the classification of certain generalized quadrangles. Our approach is novel as it utilises recently discovered links between these areas. The expected outcomes are significant progress towards these goals, as well as the development of new techniques in finite geometry.Read moreRead less
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
Normal forms and Chern-Moser connection in the study of Cauchy-Riemann Manifolds. This research project is aimed at a systematic study of Cauchy-Riemann manifolds, their holomorphic mappings and automorphisms, by means of a unifying approach based on
Chern-Moser type normal forms. The importance of Cauchy-Riemann manifolds stems from the fact that they bridge complex structure and holomorphy with the Riemannian nature of real manifolds. Construction of an analogue of the Chern-Moser normal form ....Normal forms and Chern-Moser connection in the study of Cauchy-Riemann Manifolds. This research project is aimed at a systematic study of Cauchy-Riemann manifolds, their holomorphic mappings and automorphisms, by means of a unifying approach based on
Chern-Moser type normal forms. The importance of Cauchy-Riemann manifolds stems from the fact that they bridge complex structure and holomorphy with the Riemannian nature of real manifolds. Construction of an analogue of the Chern-Moser normal form for multicodimensional Levi-nondegenerate CR-manifolds and extension of CR-mappings between them are major goals in complex analysis. Identification of Chern-Moser chains and equivariant linearisation of isotropy automorphisms are major goals in geometry.Read moreRead less
Twisted K-theory and its application to String Theory and Conformal Field Theory. String Theory is, at present, the only consistent theory of quantum gravity. Recently, twisted K-theory was proposed as the algebraic structure underlying the classification of D-branes, i.e. solitonic extended objects, in certain closed string backgrounds. In this project we aim to advance our understanding of the properties of twisted K-theory in the context of String Theory and Conformal Field Theory. The ult ....Twisted K-theory and its application to String Theory and Conformal Field Theory. String Theory is, at present, the only consistent theory of quantum gravity. Recently, twisted K-theory was proposed as the algebraic structure underlying the classification of D-branes, i.e. solitonic extended objects, in certain closed string backgrounds. In this project we aim to advance our understanding of the properties of twisted K-theory in the context of String Theory and Conformal Field Theory. The ultimate goal is to find the appropriate K-theory classifying D-branes in arbitrary closed string backgrounds or, similarly, classifying boundary Conformal Field Theories. It has already emerged that the K-theory of C*-algebras will play an important role.Read moreRead less
Dualities in String Theory and Conformal Field Theory in the context of the Geometric Langlands Program. The Langlands program ties together seemingly unrelated areas of Mathematics. Recently, in the context of the Geometric Langlands correspondence, novel connections with Theoretical Physics have emerged, thus becoming one of the most active areas of research in both Mathematics and Theoretical Physics. Australia has a number of world-renowned experts, including the two CI's, in various aspect ....Dualities in String Theory and Conformal Field Theory in the context of the Geometric Langlands Program. The Langlands program ties together seemingly unrelated areas of Mathematics. Recently, in the context of the Geometric Langlands correspondence, novel connections with Theoretical Physics have emerged, thus becoming one of the most active areas of research in both Mathematics and Theoretical Physics. Australia has a number of world-renowned experts, including the two CI's, in various aspects of the Langlands program, and is therefore well-placed to make seminal contributions. Being involved in these new developments is of crucial importance to the health of Mathematics and Theoretical Physics in Australia. An integral part of this proposal is student involvement and postgraduate training.Read moreRead less
Geometric problems from quantum theory. This Proposal is fundamental, basic research at the forefront of the application of mathematics to physical theories. The problems that will be worked on are central to much of the research activity which is presently occuring in leading centres and institutes internationally. By being a part of that research we ensure that not only is Australia involved in todays mathematical and physical advances but that we also have Australian mathematicians trained ....Geometric problems from quantum theory. This Proposal is fundamental, basic research at the forefront of the application of mathematics to physical theories. The problems that will be worked on are central to much of the research activity which is presently occuring in leading centres and institutes internationally. By being a part of that research we ensure that not only is Australia involved in todays mathematical and physical advances but that we also have Australian mathematicians trained to take advantage of the benefits those advances will bring in the future.Read moreRead less