Gravitating relativistic material bodies: A mathematical analysis. This project aims to establish the local-in-time existence and geometric uniqueness of solutions to the Einstein-Elastic equations representing systems of gravitating relativistic material bodies, and to understand the long-time behaviour of these solutions. In spite of their importance to astrophysics, almost nothing is known about the mathematical properties of solutions to the equations of motion governing gravitating systems ....Gravitating relativistic material bodies: A mathematical analysis. This project aims to establish the local-in-time existence and geometric uniqueness of solutions to the Einstein-Elastic equations representing systems of gravitating relativistic material bodies, and to understand the long-time behaviour of these solutions. In spite of their importance to astrophysics, almost nothing is known about the mathematical properties of solutions to the equations of motion governing gravitating systems of relativistic material bodies. This project would provide mathematical tools for the study of gravitating relativistic material bodies and provide guidance on developing stable numerical schemes for simulations that are essential for comparing theory with experiment. This would significantly improve current understanding of the behaviour of matter and gravitational fields near the matter-vacuum boundary of bodies and help understanding of the physics of these boundaries.Read moreRead less
Quantum invariants and hyperbolic manifolds in three-dimensional topology. The project aims to broaden our understanding of three-dimensional (3-D) spaces, including spaces that arise in engineering, microbiology and physics. It is known that all 3-D spaces can be decomposed into geometric pieces. The most common type of geometry is hyperbolic. It is also known that such spaces have algebraic structures arising from quantum physics, known as quantum invariants. Several important conjectures, bas ....Quantum invariants and hyperbolic manifolds in three-dimensional topology. The project aims to broaden our understanding of three-dimensional (3-D) spaces, including spaces that arise in engineering, microbiology and physics. It is known that all 3-D spaces can be decomposed into geometric pieces. The most common type of geometry is hyperbolic. It is also known that such spaces have algebraic structures arising from quantum physics, known as quantum invariants. Several important conjectures, based on developments in physics, assert that hyperbolic geometry and quantum invariants are deeply related, but they remain unproved. The project aims to find new relationships between hyperbolic geometry and quantum invariants, advancing our understanding of both areas.Read moreRead less
Proving the Landau-Ginzburg/Conformal Field Theory correspondence. This project aims to provide the first precise mathematical statement and geometric proof of the Landau-Ginzburg/Conformal Field Theory (LG/CFT) correspondence for simple singularities, a physically motivated principle that relates hypersurface singularities in algebraic geometry to representations of vertex algebras in conformal field theory. The formalism developed here is expected to clarify the nature of the correspondence an ....Proving the Landau-Ginzburg/Conformal Field Theory correspondence. This project aims to provide the first precise mathematical statement and geometric proof of the Landau-Ginzburg/Conformal Field Theory (LG/CFT) correspondence for simple singularities, a physically motivated principle that relates hypersurface singularities in algebraic geometry to representations of vertex algebras in conformal field theory. The formalism developed here is expected to clarify the nature of the correspondence and lead directly to generalisations beyond simple singularities, as well as provide a dictionary to translate methods of CFT into singularity theory and vice versa. These results will further cement Australia's reputation as an international leader in pure mathematics and mathematical physics research.Read moreRead less
Classical and quantum invariants of low-dimensional manifolds. This project aims to advance our understanding of knots and 3-dimensional spaces, which arise naturally in fields as diverse as physics, computer graphics, chemistry and biology. Recent ideas from quantum field theory link physics to topology in dimensions 3 and 4, leading to powerful invariants of knots and 3-dimensional manifolds that include the Jones polynomial and the 3D-index. This project aims to resolve key questions relating ....Classical and quantum invariants of low-dimensional manifolds. This project aims to advance our understanding of knots and 3-dimensional spaces, which arise naturally in fields as diverse as physics, computer graphics, chemistry and biology. Recent ideas from quantum field theory link physics to topology in dimensions 3 and 4, leading to powerful invariants of knots and 3-dimensional manifolds that include the Jones polynomial and the 3D-index. This project aims to resolve key questions relating these quantum invariants to classical topology and geometry. The project will have a major impact in low-dimensional topology, and lead to deep and unexpected connections between mathematics and mathematical physics.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
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
Discovery Early Career Researcher Award - Grant ID: DE140100633
Funder
Australian Research Council
Funding Amount
$395,169.00
Summary
Problems in the Langlands Program. The Langlands program is an international research program sitting at the interface of number theory, representation theory, algebraic geometry, and mathematical physics. The aim of this project is to prove three conjectures in this program. Settling these conjectures would lead to significant advances in the Langlands program by strengthening connections between this program and the geometry of loop groups, representations of finite groups, and representations ....Problems in the Langlands Program. The Langlands program is an international research program sitting at the interface of number theory, representation theory, algebraic geometry, and mathematical physics. The aim of this project is to prove three conjectures in this program. Settling these conjectures would lead to significant advances in the Langlands program by strengthening connections between this program and the geometry of loop groups, representations of finite groups, and representations of affine Kac-Moody algebras at the critical level.Read moreRead less
Frobenius manifolds from a geometrical and categorical viewpoint. This project aims to provide connections between Frobenius manifolds obtained from algebraic curves in diverse ways. The different constructions, using complex geometry on the one hand and category theory on the other, provide, respectively, a quantitative and qualitative view on the same Frobenius manifold. Together, these distinct points of view allow for the calculation of previously inaccessible physical quantities, and point ....Frobenius manifolds from a geometrical and categorical viewpoint. This project aims to provide connections between Frobenius manifolds obtained from algebraic curves in diverse ways. The different constructions, using complex geometry on the one hand and category theory on the other, provide, respectively, a quantitative and qualitative view on the same Frobenius manifold. Together, these distinct points of view allow for the calculation of previously inaccessible physical quantities, and point to deep new relations between algebraic, complex and differential geometry. These relations are expected to guide new fundamental research on the border of mathematics and physics.Read moreRead less
Physical realisation of enriched quantum symmetries. This project aims to investigate fundamental mathematical structures in modern category theory, providing an algebraic description of physical systems including topological order and conformal field theory. The project will study quantum symmetry, and classify and construct new classes of conformal field theories, using novel tools from enriched category theory, modular forms, and lattice gauge theory.
The main goal is to understand the lands ....Physical realisation of enriched quantum symmetries. This project aims to investigate fundamental mathematical structures in modern category theory, providing an algebraic description of physical systems including topological order and conformal field theory. The project will study quantum symmetry, and classify and construct new classes of conformal field theories, using novel tools from enriched category theory, modular forms, and lattice gauge theory.
The main goal is to understand the landscape of topological and conformal field theories, laying the foundation for new technologies based on topological order. This timely project capitalises on the recent arrival of subfactor experts in Australia, and builds capacity in mathematical research and international links in a cutting edge field.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE130100650
Funder
Australian Research Council
Funding Amount
$359,320.00
Summary
The geometry and combinatorics of moduli spaces. Moduli spaces are high-dimensional geometric objects whose rich structure holds the key to various problems in mathematics. They are of fundamental importance to modern geometry and theoretical models of the universe. This project will develop novel techniques to answer questions concerning moduli spaces and yield new insight into their structure.