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.
Discovery Early Career Researcher Award - Grant ID: DE120102657
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Group actions and K-theory: a new direction. This project investigates cutting-edge research in the mathematics of symmetries arising in nature. The aim is to significantly advance the frontiers of our knowledge by introducing new examples, original methods and a modern perspective.
Bundle gerbes: generalisations and applications. This project is fundamental, basic research at the forefront of modern differential geometry and its application to physics. It will ensure that Australia is involved in today's mathematical and physical advances and that we have Australian mathematicians trained to take advantage of the future benefits of these advances.
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
Advances in index theory. The laws of nature are often expressed in terms of differential equations, which if elliptic, have an index being the number of solutions minus the number of constraints imposed. The Atiyah-Singer Index Theorem gives a striking calculation of this index and the projects involve innovative extensions of this theory with novel applications.
Symmetry and geometric partial differential equations. This project aims to develop tools to assist the study of partial differential equations, which are fundamental to our understanding of the physical world. Symmetries of the Laplace equation are fundamental in both finding and interpreting its solutions and can be traced to the conformal symmetries of the underlying space. Only for the most symmetric of spaces, Euclidean space and the sphere, is this correspondence well understood. Using pow ....Symmetry and geometric partial differential equations. This project aims to develop tools to assist the study of partial differential equations, which are fundamental to our understanding of the physical world. Symmetries of the Laplace equation are fundamental in both finding and interpreting its solutions and can be traced to the conformal symmetries of the underlying space. Only for the most symmetric of spaces, Euclidean space and the sphere, is this correspondence well understood. Using powerful geometric tools from conformal geometry, the project will extend this to less symmetric spaces. The knowledge generated from this project will extend to more general geometric contexts providing a concrete setting for the study of the associated natural equations in curved spaces.Read moreRead less
Parametrised gauge theory and positive scalar curvature. This project aims to study innovative extensions of Seiberg-Witten gauge theory with new applications to the topology of metrics of positive scalar curvature on four-dimensional manifolds. Since Atiyah-Bott, Donaldson, Hitchin, and Seiberg-Witten’s work on various equations in gauge theory, profound applications have changed the geometry and topology of low dimensional manifolds. Parametrised index theory has obtained deep results on the t ....Parametrised gauge theory and positive scalar curvature. This project aims to study innovative extensions of Seiberg-Witten gauge theory with new applications to the topology of metrics of positive scalar curvature on four-dimensional manifolds. Since Atiyah-Bott, Donaldson, Hitchin, and Seiberg-Witten’s work on various equations in gauge theory, profound applications have changed the geometry and topology of low dimensional manifolds. Parametrised index theory has obtained deep results on the topology of metrics of positive scalar curvature in higher dimensions, but these methods do not work in the case of the fourth dimension. This project will develop (parametrised) Seiberg-Witten gauge theory as a new approach to the study of the topology of metrics of positive scalar curvature in four dimensions. Expected outcomes include new invariants related to positive scalar curvature in four dimensions.Read moreRead less
Symmetries in real and complex geometry. This project concerns an important area of abstract modern geometry. The results and techniques of the project will lead to significant progress in this area. It will benefit the national scientific reputation, strengthen the research profile of the home institutions, and provide training to young researchers.
Homotopical structures in algebraic, analytic, and equivariant geometry. This is a project for fundamental research in pure mathematics. It is focused on an emerging subfield of complex geometry concerned with spaces and maps that exhibit exceptional flexibility properties, which often go hand-in-hand with a high degree of symmetry. The project aims to develop the foundations of this new area, solve several open problems, and pursue interconnections with and applications to algebraic geometry, c ....Homotopical structures in algebraic, analytic, and equivariant geometry. This is a project for fundamental research in pure mathematics. It is focused on an emerging subfield of complex geometry concerned with spaces and maps that exhibit exceptional flexibility properties, which often go hand-in-hand with a high degree of symmetry. The project aims to develop the foundations of this new area, solve several open problems, and pursue interconnections with and applications to algebraic geometry, complex analysis, geometric invariant theory, and topology.Read moreRead less
Twisted K-theory, higher geometry and operator algebras. This project aims to develop new theory and techniques linking twisted K-theory, higher-geometry and operator algebras. These are all fundamental areas of mathematics with applications both within mathematics itself and to mathematical physics, particularly in string theory. Anticipated outcomes are fundamental advances in knowledge in mathematics and mathematical physics, enhancement of Australia's international mathematical reputation an ....Twisted K-theory, higher geometry and operator algebras. This project aims to develop new theory and techniques linking twisted K-theory, higher-geometry and operator algebras. These are all fundamental areas of mathematics with applications both within mathematics itself and to mathematical physics, particularly in string theory. Anticipated outcomes are fundamental advances in knowledge in mathematics and mathematical physics, enhancement of Australia's international mathematical reputation and collaborative linkages, and the training of the next generation of Australian mathematicians.Read moreRead less