Ubiquity of K-theory and T-duality. An abstract mathematical tool, called K-theory, has recently found application in two, not obviously related, areas of physics: the classification of D-branes in String Theory, and topological phases in Condensed Matter Theory. This project aims to advance the development of K-theory using ideas from physics. In particular, the project aims to generalise previous constructions, such as T-duality, to loop spaces, and to develop the K-theory relevant to the clas ....Ubiquity of K-theory and T-duality. An abstract mathematical tool, called K-theory, has recently found application in two, not obviously related, areas of physics: the classification of D-branes in String Theory, and topological phases in Condensed Matter Theory. This project aims to advance the development of K-theory using ideas from physics. In particular, the project aims to generalise previous constructions, such as T-duality, to loop spaces, and to develop the K-theory relevant to the classification of topological phases in strongly interacting systems. This project involves postgraduate training as a crucial tool in achieving its aims and enhances Australia's position at the forefront of international research.Read moreRead less
Quadratic fusion categories: A frontier in subfactor theory. This project aims to investigate the quantum symmetries of the quadratic fusion categories. Fusion categories are mathematical structures that generalise the symmetries of finite groups. These structures arise as invariants of subfactors in operator algebras and in mathematical models of conformal field theory. The quadratic fusion categories encompass most known subfactors that do not come from finite or quantum groups and form a vast ....Quadratic fusion categories: A frontier in subfactor theory. This project aims to investigate the quantum symmetries of the quadratic fusion categories. Fusion categories are mathematical structures that generalise the symmetries of finite groups. These structures arise as invariants of subfactors in operator algebras and in mathematical models of conformal field theory. The quadratic fusion categories encompass most known subfactors that do not come from finite or quantum groups and form a vast frontier about which little is known. By uncovering the symmetries of the quadratic fusion categories, the project will advance subfactor theory and provide new models for conformal field theory. Progress in these fields will have applications to the emerging technology of quantum computing.Read moreRead less
Symmetries of subfactors. A subfactor is a mathematical object that encodes "quantum" symmetries which may be thought of as generalisations of group symmetries. This project will study subfactors and classify families of subfactor symmetries which include the exotic subfactors of small index. It will also develop computational tools for analysing and cataloguing these symmetries. This project contributes to the development of operator algebra theory, and the new mathematical fields of quantum al ....Symmetries of subfactors. A subfactor is a mathematical object that encodes "quantum" symmetries which may be thought of as generalisations of group symmetries. This project will study subfactors and classify families of subfactor symmetries which include the exotic subfactors of small index. It will also develop computational tools for analysing and cataloguing these symmetries. This project contributes to the development of operator algebra theory, and the new mathematical fields of quantum algebra and quantum topology; it also has applications to physical models.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
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
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.
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.
Cohomology, symbolic dynamics and operator algebras. Operator algebras encode a kind of virtual space which is very different from the visible three-dimensional world. This is the arena of quantum mechanics. This project will adapt the tools of classical topology - the study of space and shape - to probe the structure of virtual space and glean new insights into its peculiar properties.
Taming infinite dimensions: quasidiagonality and nuclear dimension. This project aims to develop new methods for understanding regularity properties of operator algebras. These play a crucial role in the development of quantum physics, quantum computing and in topological insulators. Operator algebras constitute the mathematical underpinnings of quantum mechanics. This project aims to analyse nuclear dimension and quasidiagonality of operator algebras: two recently developed and exceptionally im ....Taming infinite dimensions: quasidiagonality and nuclear dimension. This project aims to develop new methods for understanding regularity properties of operator algebras. These play a crucial role in the development of quantum physics, quantum computing and in topological insulators. Operator algebras constitute the mathematical underpinnings of quantum mechanics. This project aims to analyse nuclear dimension and quasidiagonality of operator algebras: two recently developed and exceptionally important regularity properties. This should deliver significant benefits, including an enhanced understanding of operator algebras and strengthened research capacity and standing for Australia.Read moreRead less
Operator algebras as models for dynamics and geometry. Operator algebra is the mathematical theory which describes quantum physics and predicts how quantum systems will behave. Through this project, the researcher's recent discoveries in operator algebra will give us new insight into the dynamics and geometry - that is, the behaviour and shape - of the quantum world.