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
Trisections, triangulations and the complexity of manifolds. This project aims at practical representations of 3-dimensional and 4-dimensional spaces as needed in applications. Topology is the mathematical study of the shapes of spaces. Geometry endows spaces with additional structure such as distance, angle and curvature. Special combinatorial structures, such as minimal triangulations, are often closely connected to geometric structures or topological properties. This project aims to construct ....Trisections, triangulations and the complexity of manifolds. This project aims at practical representations of 3-dimensional and 4-dimensional spaces as needed in applications. Topology is the mathematical study of the shapes of spaces. Geometry endows spaces with additional structure such as distance, angle and curvature. Special combinatorial structures, such as minimal triangulations, are often closely connected to geometric structures or topological properties. This project aims to construct computable invariants, connectivity results for triangulations, and algorithms to recognise fundamental topological properties and structures such as trisections and bundles.Read moreRead less
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.
Geometric transforms and duality. This Proposal 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.
Australian Laureate Fellowships - Grant ID: FL170100020
Funder
Australian Research Council
Funding Amount
$1,638,060.00
Summary
Advances in index theory and applications. The project aims to develop novel techniques to investigate Geometric analysis on infinite dimensional bundles, as well as Geometric analysis of pathological spaces with Cantor set as fibre, that arise in models for the fractional quantum Hall effect and topological matter, areas recognised with the 1998 and 2016 Nobel Prizes. Building on the applicant’s expertise in the area, the project will involve postgraduate and postdoctoral training in order to e ....Advances in index theory and applications. The project aims to develop novel techniques to investigate Geometric analysis on infinite dimensional bundles, as well as Geometric analysis of pathological spaces with Cantor set as fibre, that arise in models for the fractional quantum Hall effect and topological matter, areas recognised with the 1998 and 2016 Nobel Prizes. Building on the applicant’s expertise in the area, the project will involve postgraduate and postdoctoral training in order to enhance Australia’s position at the forefront of international research in Geometric Analysis. Ultimately, the project will enhance Australia's leading position in the area of Index Theory by developing novel techniques to solve challenging conjectures, and mentoring HDR students and ECRs.Read moreRead less
Geometric analysis on non-compact and singular spaces. The project will involve mathematical research of international stature in an effervescent field: geometric analysis on singular spaces. Different points of view will be used, stimulating in this way fruitful interactions between analysis and geometry which will lead to striking new relationships as well as implications in physics and engineering.
Curvature flows and spectral estimates. Curvature flows are a class of geometrically motivated equations, modelled on the heat equation. Recently, researchers have developed new methods for studying the regularity of solutions to these equations, and applied them to a different problem, that of estimating quantities depending on the smaller eigenvalues of a Schroedinger operator. This project builds on the early success of this research and will produce a new understanding of the behaviour of ei ....Curvature flows and spectral estimates. Curvature flows are a class of geometrically motivated equations, modelled on the heat equation. Recently, researchers have developed new methods for studying the regularity of solutions to these equations, and applied them to a different problem, that of estimating quantities depending on the smaller eigenvalues of a Schroedinger operator. This project builds on the early success of this research and will produce a new understanding of the behaviour of eigenvalues, establish sharp estimates for spectral quantities, particularly on manifolds with curvature bounds, and find optimal conditions under which non-compact solutions to curvature flows are stable.Read moreRead less
Heat kernel and Riesz transform on non-compact metric measure spaces. This project will develop new techniques in heat kernel theory, with applications to such important topics as Schrodinger model for quantum mechanics. The proposed research is at the forefront of research in harmonic analysis and partial differential equations and will further enhance Australia's high international standing in these research fields.
Microlocal Analysis - A Unified Approach for Geometric Models in Biology . This project will use microlocal analysis to create a unified approach for predicting the outcome of a broad class of diffusion and reaction-diffusion models. This will replace the traditional theory which is no longer adequate for the level of geometric complexity demanded of current models arising in biology/ecology. This project will address the urgent need for a systematic theoretical underpinning of diffusion/reacti .... Microlocal Analysis - A Unified Approach for Geometric Models in Biology . This project will use microlocal analysis to create a unified approach for predicting the outcome of a broad class of diffusion and reaction-diffusion models. This will replace the traditional theory which is no longer adequate for the level of geometric complexity demanded of current models arising in biology/ecology. This project will address the urgent need for a systematic theoretical underpinning of diffusion/reaction-diffusion in geometric settings whose scope of application is broader than the the existing patchwork of methods.
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Discovery Early Career Researcher Award - Grant ID: DE190101063
Funder
Australian Research Council
Funding Amount
$319,474.00
Summary
Geometric flows and distinguished geometric structures with symmetry. This project aims to carry out analysis for the Ricci flow and a number of recent flows in complex geometry. Geometric evolution equations are a trending topic in modern mathematics, with successful applications such as the celebrated proof of the century-old Poincaré Conjecture in 2002-03 by means of the Ricci flow. An analysis of the behaviour of solutions with symmetry is an essential component in any mature theory on the s ....Geometric flows and distinguished geometric structures with symmetry. This project aims to carry out analysis for the Ricci flow and a number of recent flows in complex geometry. Geometric evolution equations are a trending topic in modern mathematics, with successful applications such as the celebrated proof of the century-old Poincaré Conjecture in 2002-03 by means of the Ricci flow. An analysis of the behaviour of solutions with symmetry is an essential component in any mature theory on the subject. The project will use the novel tools and ideas resulting from this analysis to address the Alekseevskii Conjecture, a 40-year old fundamental open question in the field with major ramifications in mathematics and beyond. This project will further reinforce Australia's leading role in geometric analysis.Read moreRead less