Universal structures in stringy extra dimensions. The project aims to study properties of extra dimensions in string theory by means of techniques from supersymmetric gauge theory. This new approach makes it possible to study areas in the landscape of stringy extra dimensions that have not been accessible before. The project expects to uncover new universal features. This will have significant impact on string theory and mathematics. Expected outcomes of this project include answers to conceptua ....Universal structures in stringy extra dimensions. The project aims to study properties of extra dimensions in string theory by means of techniques from supersymmetric gauge theory. This new approach makes it possible to study areas in the landscape of stringy extra dimensions that have not been accessible before. The project expects to uncover new universal features. This will have significant impact on string theory and mathematics. Expected outcomes of this project include answers to conceptual questions in string theory, new types of extra dimensions, and new methods to compute quantum corrections in string theory. This should provide significant benefits, such as interdisciplinary collaborations at the national and international level and a strengthening of string theory in Australia.Read moreRead less
The Ricci curvature of homogeneous spaces. The geometry of homogeneous spaces is an area of research with applications in numerous fields, including topology, harmonic analysis, relativity and quantum theory. This project aims to resolve a fundamental problem in this area, known as the prescribed Ricci curvature problem for homogeneous metrics, and to settle the important and closely related question of Ricci iteration existence and convergence. Moreover, the project aims to exploit the interpla ....The Ricci curvature of homogeneous spaces. The geometry of homogeneous spaces is an area of research with applications in numerous fields, including topology, harmonic analysis, relativity and quantum theory. This project aims to resolve a fundamental problem in this area, known as the prescribed Ricci curvature problem for homogeneous metrics, and to settle the important and closely related question of Ricci iteration existence and convergence. Moreover, the project aims to exploit the interplay between geometry and algebra to provide new insight into the physically significant problem of classifying unitary Lie algebra representations. This project is expected to facilitate interdisciplinary interaction leading to exciting developments across a range of fields.Read moreRead less
Lie superalgebra representations: a geometric approach. The concept of a Lie group provides a mathematical underpinning for the idea of symmetry in mathematics, physics and chemistry. The project aims to advance two fundamental problems related to this concept: classification of unitary representations of Lie superalgebras, and the prescribed Ricci curvature problem on Lie groups. The research builds on newly-discovered connections between these problems to achieve exciting progress in their res ....Lie superalgebra representations: a geometric approach. The concept of a Lie group provides a mathematical underpinning for the idea of symmetry in mathematics, physics and chemistry. The project aims to advance two fundamental problems related to this concept: classification of unitary representations of Lie superalgebras, and the prescribed Ricci curvature problem on Lie groups. The research builds on newly-discovered connections between these problems to achieve exciting progress in their resolution. Outcomes are expected to find applications across a range of fields, such as condensed matter physics, particle physics, quantum field theory and knot theory. Anticipated benefits include stronger links between different areas of science achieved through a deeper understanding of symmetry.Read moreRead less
Multi-dimensionally consistent integrable systems in geometry and algebra. This project aims to address in an innovative manner a long-standing open problem in nonlinear mathematics, namely the determination of the algebraic and geometric origin of integrable systems. It is expected to make a fundamental contribution towards integrable systems theory. The latter provides unique access to the analytic treatment of nonlinear phenomena not only in physics but also a remarkably diverse range of area ....Multi-dimensionally consistent integrable systems in geometry and algebra. This project aims to address in an innovative manner a long-standing open problem in nonlinear mathematics, namely the determination of the algebraic and geometric origin of integrable systems. It is expected to make a fundamental contribution towards integrable systems theory. The latter provides unique access to the analytic treatment of nonlinear phenomena not only in physics but also a remarkably diverse range of areas in mathematics. Expected outcomes include extended, unified and novel key mathematical concepts in a discrete setting and their applications in algebraic and geometric contexts. Due to the choice of participants, it is anticipated that Australia will benefit from strengthened research collaborations with Germany.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
Geometric themes in the theory of Lie supergroups and their quantisations. This project aims to develop mathematics on the geometry of super spaces and the algebra of super transformations, which are the cornerstones of the mathematical foundation of supersymmetry. The Large Hadron Collider at the European Organization for Nuclear Research is investigating supersymmetry as a possible symmetry of fundamental physics. Its empirical verification would confirm the existence of new constituents of ma ....Geometric themes in the theory of Lie supergroups and their quantisations. This project aims to develop mathematics on the geometry of super spaces and the algebra of super transformations, which are the cornerstones of the mathematical foundation of supersymmetry. The Large Hadron Collider at the European Organization for Nuclear Research is investigating supersymmetry as a possible symmetry of fundamental physics. Its empirical verification would confirm the existence of new constituents of matter, and reveal deep structures of space-time beyond the framework of Einstein's general relativity. Results of the project are expected to be directly applicable to high energy physics.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE220100918
Funder
Australian Research Council
Funding Amount
$426,000.00
Summary
Teichmueller dynamics and the birational geometry of moduli space. The project aims to leverage recent cutting-edge advances in the area of Teichmueller dynamics to answer longstanding open questions from algebraic geometry on the moduli space of curves, an object with deep connections to many diverse areas of science including quantum gravity and theoretical physics. The project expects to generate new theories and increased understanding in both areas through the innovation of relating these p ....Teichmueller dynamics and the birational geometry of moduli space. The project aims to leverage recent cutting-edge advances in the area of Teichmueller dynamics to answer longstanding open questions from algebraic geometry on the moduli space of curves, an object with deep connections to many diverse areas of science including quantum gravity and theoretical physics. The project expects to generate new theories and increased understanding in both areas through the innovation of relating these perspectives, as well as uncovering new connections between the viewpoints. Further benefits should include building international collaborations and the contribution of this diverse perspective to the growing algebraic geometry community in Australia and to mathematics and related scientific fields more generally.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE140100259
Funder
Australian Research Council
Funding Amount
$379,549.00
Summary
New Compactifications of Moduli Spaces of Curves. A moduli space is a geometric representation of a system of algebraic equations. Many fundamental problems in algebra, dynamics and physics can be reformulated as questions concerning the geometry of moduli spaces, in particular the moduli space of curves. This project will produce a systematic classification of compactifications of the moduli space of curves, and develop new tools for investigating the geometry of these compactifications. One of ....New Compactifications of Moduli Spaces of Curves. A moduli space is a geometric representation of a system of algebraic equations. Many fundamental problems in algebra, dynamics and physics can be reformulated as questions concerning the geometry of moduli spaces, in particular the moduli space of curves. This project will produce a systematic classification of compactifications of the moduli space of curves, and develop new tools for investigating the geometry of these compactifications. One of these compactifications already lies at the centre of an extraordinary web of connections linking together topology, combinatorics and quantum field theory, and there is a strong possibility that this web fits into a more comprehensive picture involving all compactifications on an equal footing.Read moreRead less