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
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
Comprehensive Study of Kahler-Ricci Flows. The intended outcome of the project is to reveal the deep relation between geometry and topology of underlying spaces. Ricci flow has attracted major attention in pure mathematics over the past 30 years, including ground-breaking contributions by Perelman on Ricci flow regarding the famous Poincare and Thurston's Geometrisation Conjectures. The project focuses on the complex version of Ricci flow: Kahler-Ricci flow. The project plans to explore the Kahl ....Comprehensive Study of Kahler-Ricci Flows. The intended outcome of the project is to reveal the deep relation between geometry and topology of underlying spaces. Ricci flow has attracted major attention in pure mathematics over the past 30 years, including ground-breaking contributions by Perelman on Ricci flow regarding the famous Poincare and Thurston's Geometrisation Conjectures. The project focuses on the complex version of Ricci flow: Kahler-Ricci flow. The project plans to explore the Kahler-Ricci flow in the closed and complete non-compact settings and the corresponding versions of Geometric Minimal Model Program; and the Kahler-Ricci flow in the Fano manifold setting and stability conditions.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE180101348
Funder
Australian Research Council
Funding Amount
$328,075.00
Summary
Singularity analysis for ricci flow and mean curvature flow. This project aims to investigate the central problem of singularity formation in Ricci flow and mean-curvature flow by profiling singular solutions and determining their stability and genericity. Geometric flows are powerful and successful ways of understanding classical problems in geometry and topology with applications in disciplines such as materials science and medical imaging. This project will generate significant results in sin ....Singularity analysis for ricci flow and mean curvature flow. This project aims to investigate the central problem of singularity formation in Ricci flow and mean-curvature flow by profiling singular solutions and determining their stability and genericity. Geometric flows are powerful and successful ways of understanding classical problems in geometry and topology with applications in disciplines such as materials science and medical imaging. This project will generate significant results in singularity analysis and will enrich understanding of geometric flows at and past singularities, deepen the theory of geometric flows, and enhance their applications in mathematics and science.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
Algebraic Schubert geometry and unitary reflection groups. This project aims to generalise the recent work of Elias and Williamson to the complex case. Fundamental to the study of symmetry are the ubiquitous Coxeter groups, which have an associated set of critically important ‘Kazhdan-Lusztig polynomials’. For some Coxeter groups, these may be interpreted in terms of classical geometry, leading to deep positivity properties for their coefficients. Elias and Williamson have recently shown that th ....Algebraic Schubert geometry and unitary reflection groups. This project aims to generalise the recent work of Elias and Williamson to the complex case. Fundamental to the study of symmetry are the ubiquitous Coxeter groups, which have an associated set of critically important ‘Kazhdan-Lusztig polynomials’. For some Coxeter groups, these may be interpreted in terms of classical geometry, leading to deep positivity properties for their coefficients. Elias and Williamson have recently shown that this geometry may be simulated algebraically for any Coxeter group, so positivity for Kazhdan-Lusztig polynomials holds for all Coxeter groups. This result has explosive consequences in many areas of geometry and algebra. This project is designed to extend these results to complex unitary reflection groups, with potentially dramatic consequences in number theory, representation theory and topology.Read moreRead less
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
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
Discovery Early Career Researcher Award - Grant ID: DE200101834
Funder
Australian Research Council
Funding Amount
$418,410.00
Summary
The structure of singularities in geometric flows. The proposed research aims to develop our understanding of the structure of singularities in mean curvature and related flows, with certain applications in mind.