Holonomy groups in Lorentzian geometry. The project studies mathematical models used in physical theories, such as general relativity and string theory, to create a global picture of the universe. The outcomes will enhance the role Australia plays in these developments and contribute to the mathematical knowledge which lies at the foundations of modern technologies.
Symmetric functions and Hodge polynomials. This project aims to explain a connection between two seemingly disparate mathematical notions: mixed Hodge polynomials of certain varieties, naturally arising in algebraic geometry, and Macdonald polynomials from the theory of symmetric functions. This project will resolve this connection using symmetric function theory, algebraic combinatorics and representation theory. This project could enhance Australia's international reputation in algebraic combi ....Symmetric functions and Hodge polynomials. This project aims to explain a connection between two seemingly disparate mathematical notions: mixed Hodge polynomials of certain varieties, naturally arising in algebraic geometry, and Macdonald polynomials from the theory of symmetric functions. This project will resolve this connection using symmetric function theory, algebraic combinatorics and representation theory. This project could enhance Australia's international reputation in algebraic combinatorics, combinatorial representation theory and algebraic geometry.Read moreRead less
Modular character sheaves. This project aims to complete the fundamental mathematical theory of modular group representations, the algebraic description of symmetry over finite number systems. Group representation theory can be applied to any linear problem involving symmetry. However, the modular case, where the characteristic of the underlying field is a prime number, is less understood than real or complex scalars, and this lack of understanding blocks potential applications. This project wil ....Modular character sheaves. This project aims to complete the fundamental mathematical theory of modular group representations, the algebraic description of symmetry over finite number systems. Group representation theory can be applied to any linear problem involving symmetry. However, the modular case, where the characteristic of the underlying field is a prime number, is less understood than real or complex scalars, and this lack of understanding blocks potential applications. This project will use geometric methods to answer questions about modular representations of the finite groups of Lie type, the most important class of finite groups. This project could make modular representation theory essential for computations, enabling faster solutions to problems of linear algebra and allowing future applications in such areas as data transmission technology.Read moreRead less
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
Topology in seven dimensions. Aims: The project aims to give a complete classification of a certain class of 7-dimensional spaces; namely simply-connected spin 7-manifolds. We also present related programs classify G_2-structures on 7-manifolds.
Significance: the proposed classification will be a signature achievement in the topology of manifolds, with applications likely in both geometry and mathematical physics.
Expected outcomes: The project will produce a series of papers published in high ....Topology in seven dimensions. Aims: The project aims to give a complete classification of a certain class of 7-dimensional spaces; namely simply-connected spin 7-manifolds. We also present related programs classify G_2-structures on 7-manifolds.
Significance: the proposed classification will be a signature achievement in the topology of manifolds, with applications likely in both geometry and mathematical physics.
Expected outcomes: The project will produce a series of papers published in high quality journals and enhanced scientific collaboration between Australia and the United Kingdom.
Benefits: The project will enhance Australia's research reputation by producing excellent research in a field not historically represented in the country.Read moreRead less
Symmetry and geometric structures. This is a fundamental research project in mathematics, especially concerned with the interaction between symmetry, differential equations, and geometry. Based on many classical and recently discovered instances, the aim of the project is to use symmetries to build and understand curved geometric structures from their flat counterparts.
Higher order curvature flow of curves and hypersurfaces. This project aims to analyse higher order geometric partial differential equations that have important mathematical applications in differential geometry of submanifolds as well as practical applications in physics and mathematical biology. The project aims to prove new general principles that reveal properties of these higher order elliptic and parabolic partial differential equations, producing a unified framework with applications to va ....Higher order curvature flow of curves and hypersurfaces. This project aims to analyse higher order geometric partial differential equations that have important mathematical applications in differential geometry of submanifolds as well as practical applications in physics and mathematical biology. The project aims to prove new general principles that reveal properties of these higher order elliptic and parabolic partial differential equations, producing a unified framework with applications to various specific problems. This project aims to increase Australia's research capacity in geometric evolution problems, provide training for some of Australia's next generation of mathematicians and build Australia's international reputation for significant research in geometric analysis.Read moreRead less
Moduli spaces of geometric structures. One of the most spectacular recent advances in mathematics was Perelman's resolution of Thurston's geometrisation program for three-dimensional spaces. This makes it very important to understand sets of geometric structures on such spaces, called moduli spaces. The challenge of this project is to develop practical methods to compute moduli spaces, understand their global properties, and use them to define new invariants giving insights about shape, geometry ....Moduli spaces of geometric structures. One of the most spectacular recent advances in mathematics was Perelman's resolution of Thurston's geometrisation program for three-dimensional spaces. This makes it very important to understand sets of geometric structures on such spaces, called moduli spaces. The challenge of this project is to develop practical methods to compute moduli spaces, understand their global properties, and use them to define new invariants giving insights about shape, geometry and algebra of a space. This is a key problem as moduli spaces describe whether a space is rigid or can be deformed. They are a central object in several fields of mathematics, including geometry and topology, gauge theory, dynamical systems, mathematical physics and invariant theory.Read moreRead less
Invariants, geometric and discrete structures on manifolds. This project aims to develop practical methods for finding geometric and discrete structures on manifolds in both low and high dimensions and advancing our understanding of the information that physics is providing about these spaces. Recently there have been spectacular advances in understanding 3-D spaces and the interaction between ideas in mathematical physics (quantum invariants, string theory) and such spaces. In this project, the ....Invariants, geometric and discrete structures on manifolds. This project aims to develop practical methods for finding geometric and discrete structures on manifolds in both low and high dimensions and advancing our understanding of the information that physics is providing about these spaces. Recently there have been spectacular advances in understanding 3-D spaces and the interaction between ideas in mathematical physics (quantum invariants, string theory) and such spaces. In this project, the first aim is to construct structures with good geometric properties on 3- and 4-manifolds, using triangulations. The second aim is to study combinatorial decompositions of n-manifolds, using our new technique of multisections and also searching for polyhedral metrics of non-positive curvature. The third aim is to connect quantum invariants and geometric structures, again using triangulations.Read moreRead less
Fully nonlinear elliptic equations and applications. This project aims to develop new methods to solve challenging problems in fully nonlinear elliptic equations, and to confirm and enhance Australia as a world leader in this very active area. In addition to high impact publications, this highly innovative research also provides continued building of expertise and training in the area.