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
Equilibrium states and fine structure for operator algebras. This project is in pure mathematics, in the broad area of functional analysis, and focuses specifically on operator algebras. Kubo-Martin-Schwinger (KMS) states on operator algebras encode equilibria of C*-algebraic dynamical systems. This project aims to take a novel view of KMS data as a repository of fine operator-algebraic structure. It aims to develop a theory whereby KMS states recover structural details like primitive-ideal stru ....Equilibrium states and fine structure for operator algebras. This project is in pure mathematics, in the broad area of functional analysis, and focuses specifically on operator algebras. Kubo-Martin-Schwinger (KMS) states on operator algebras encode equilibria of C*-algebraic dynamical systems. This project aims to take a novel view of KMS data as a repository of fine operator-algebraic structure. It aims to develop a theory whereby KMS states recover structural details like primitive-ideal structure and simplicity. The project is expected to determine to what extent the KMS simplex of combinatorial operator algebra remembers underlying combinatorial data. It also aims to explore KMS states on combinatorial operator algebras as a new point of interaction between the two main branches of modern operator-algebra theory.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
Schur decompositions and related problems in operator theory. This project aims to solve some famous problems concerning eigenvalue decompositions in operator theory through new collaborations and by connecting new areas of mathematics. Eigenvalue decomposition is a central concept in mathematics with many applications in science and engineering. One hundred years since its development, however, it is still not known how to decompose certain important operators that arise in analysis and geometr ....Schur decompositions and related problems in operator theory. This project aims to solve some famous problems concerning eigenvalue decompositions in operator theory through new collaborations and by connecting new areas of mathematics. Eigenvalue decomposition is a central concept in mathematics with many applications in science and engineering. One hundred years since its development, however, it is still not known how to decompose certain important operators that arise in analysis and geometry. The project is expected to provide new technology to achieve this, promising new understanding and new applications.Read moreRead less
Structured barrier and penalty functions in infinite dimensional optimisation and analysis. Very large scale tightly-constrained optimisation problems are ubiquitous and include water management, traffic flow, and imaging at telescopes and hospitals. Massively parallel computers can solve such problems and provide physically realisable solution only if subtle design issues are mastered. Resolving such issues is the goal of this project.
The boundaries of index theory. In recent years there has been an influx of new ideas from other disciplines into mathematics and this has led to major advances in many areas, notably geometry and topology. Classical problems have been solved and new perspectives exposed. In this spirit this project will use the methods of noncommutative analysis and noncommutative geometry to extend the mathematical area of spectral geometry. A primary objective is to determine how the geometric and differentia ....The boundaries of index theory. In recent years there has been an influx of new ideas from other disciplines into mathematics and this has led to major advances in many areas, notably geometry and topology. Classical problems have been solved and new perspectives exposed. In this spirit this project will use the methods of noncommutative analysis and noncommutative geometry to extend the mathematical area of spectral geometry. A primary objective is to determine how the geometric and differential structure of certain spaces interacts with the new spectral invariants that will be introduced. The project aims to obtain more subtle and refined information about these spaces. In this fashion it expects to resolve several long standing questions in mathematics. Read moreRead less
Groupoids as bridges between algebra and analysis. This pure mathematics project focuses on the interplay between abstract algebra and the area of functional analysis known as operator algebras. Specifically, it is intended to deal with generalisations of graph C*-algebras and of Leavitt path algebras. Over the last decade, researchers have discovered striking similarities between these areas, but no unifying result that would allow them to transfer techniques and theorems systematically from on ....Groupoids as bridges between algebra and analysis. This pure mathematics project focuses on the interplay between abstract algebra and the area of functional analysis known as operator algebras. Specifically, it is intended to deal with generalisations of graph C*-algebras and of Leavitt path algebras. Over the last decade, researchers have discovered striking similarities between these areas, but no unifying result that would allow them to transfer techniques and theorems systematically from one to the other. Recent research suggests that groupoid models for both algebras and C*-algebras may provide the missing link. This project aims to determine the role of groupoids in the two theories, and analyse and exploit the resulting synergies between abstract algebra and operator algebras.Read moreRead less
Graded K-theory as invariants for path algebras. This pure mathematics project focuses on Leavitt path algebras, which are structures that naturally arise from movements on directed graphs. These algebras appear in diverse areas (eg analysis, noncommutative geometry, representation theory and group theory). The aim of this project is to understand the behaviour of Leavitt path algebras and to classify them completely by means of graded K-theory. The project is an algebraic counterpart to graph C ....Graded K-theory as invariants for path algebras. This pure mathematics project focuses on Leavitt path algebras, which are structures that naturally arise from movements on directed graphs. These algebras appear in diverse areas (eg analysis, noncommutative geometry, representation theory and group theory). The aim of this project is to understand the behaviour of Leavitt path algebras and to classify them completely by means of graded K-theory. The project is an algebraic counterpart to graph C*-algebras (analytic structures that originated in Australian universities); both subjects have become areas of intensive research globally. The expected outcomes are to classify Leavitt path algebras, and to find a bridge (via graded K-theory) to graph C*-algebras and symbolic dynamics.Read moreRead less
Analysis of nonlinear partial differential equations describing singular phenomena. This project will advance knowledge on a huge variety of systems with applications across the sciences by providing new methods to investigate nonlinear partial differential equations with singularities. The analysis of many models describing physical and biological systems relies on such equations.
Singular solutions for nonlinear elliptic and parabolic equations. The analysis of many models fundamental to physical and biological sciences is obstructed by singularities. This project aims to discover and classify the singular solutions for two important types of nonlinear equations: elliptic and parabolic. The project expects to generate novel methods to decipher singularities by using innovative approaches from geometric analysis and dynamical systems. Expected outcomes of this project inc ....Singular solutions for nonlinear elliptic and parabolic equations. The analysis of many models fundamental to physical and biological sciences is obstructed by singularities. This project aims to discover and classify the singular solutions for two important types of nonlinear equations: elliptic and parabolic. The project expects to generate novel methods to decipher singularities by using innovative approaches from geometric analysis and dynamical systems. Expected outcomes of this project include new and powerful tools to advance a more general theory of singularities. This should provide significant benefits, such as new mathematical knowledge on key issues on singularities lying at the forefront of international research and enhanced expertise in an area of worldwide recognition for Australia.Read moreRead less