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
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.
Higher dimensional methods for algebras and dynamical systems. Australian researchers have pioneered recent research in combinatorial C*-algebras. We are now uniquely placed to capitalise on this situation to make significant connections with research in dynamical systems. This project will thus position Australian mathematics at the nexus of two important research areas.
Harmonic analysis of rough oscillations. This project intends to explore new perspectives in harmonic analysis. Harmonic analysis is a set of mathematical techniques used in many branches of science and engineering to analyse complex signals (functions). It is highly effective in modelling phenomena such as the propagation of electromagnetic waves, but it is currently limited to propagation occurring in a simple-enough medium. An intense international research effort in harmonic analysis is curr ....Harmonic analysis of rough oscillations. This project intends to explore new perspectives in harmonic analysis. Harmonic analysis is a set of mathematical techniques used in many branches of science and engineering to analyse complex signals (functions). It is highly effective in modelling phenomena such as the propagation of electromagnetic waves, but it is currently limited to propagation occurring in a simple-enough medium. An intense international research effort in harmonic analysis is currently under way to lift this limitation. This project is part of that effort, and aims to unite two of its fundamental directions of development: one focusing on the roughness of the medium; and one focusing on the interaction between highly oscillatory aspects of the function and the geometry of the medium.Read moreRead less
Noncommutative probability and analysis. This project contributes to the development of new mathematical disciplines, noncommutative and free probability theories, which first appeared in the last 20 years and which is expected to have important applications in quantum mechanics and hence electronics and computing.