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.
Geometry in projection methods and fixed-point theory. This project aims to resolve mathematical challenges arising from problems with specific structure typical for key modern applications, such as big data optimisation, chemical engineering and medical imaging. We focus on developing new mathematical tools for the analysis of projection methods and accompanying fixed point theory, specifically targeting the refinement of the geometric intuition for algorithm design techniques to inform the imp ....Geometry in projection methods and fixed-point theory. This project aims to resolve mathematical challenges arising from problems with specific structure typical for key modern applications, such as big data optimisation, chemical engineering and medical imaging. We focus on developing new mathematical tools for the analysis of projection methods and accompanying fixed point theory, specifically targeting the refinement of the geometric intuition for algorithm design techniques to inform the implementation of optimal methods for huge-scale optimisation problems.Read moreRead less
There and back again: operator algebras, algebras and dynamical systems. The aim of this project is to develop mathematics that enables us to transfer information back and forth between dynamical systems and algebras, including operator algebras. Dynamical systems - systems that change over time - are ubiquitous, and central to modern mathematics and its applications. In mathematics, dualities allow us to translate questions from one context to another in which they are easier to solve and then ....There and back again: operator algebras, algebras and dynamical systems. The aim of this project is to develop mathematics that enables us to transfer information back and forth between dynamical systems and algebras, including operator algebras. Dynamical systems - systems that change over time - are ubiquitous, and central to modern mathematics and its applications. In mathematics, dualities allow us to translate questions from one context to another in which they are easier to solve and then translate the answer back again. Expected outcomes include increased understanding of the relationship between operator algebras and the dynamical systems that they represent. Benefits include enhanced international collaboration, and increased Australian capacity in pure mathematics, particularly operator algebras.Read moreRead less
From actions to operator algebras and their equilibrium states. This project aims to construct C*-algebras from various types of actions and analyse their equilibrium states. Operator algebras are widely used in mathematics and to describe physical systems. They are technically challenging to work with and impossible to fully classify, making detailed analysis of large classes of examples important research in the area. This project will construct C*-algebras from various actions; analyse their ....From actions to operator algebras and their equilibrium states. This project aims to construct C*-algebras from various types of actions and analyse their equilibrium states. Operator algebras are widely used in mathematics and to describe physical systems. They are technically challenging to work with and impossible to fully classify, making detailed analysis of large classes of examples important research in the area. This project will construct C*-algebras from various actions; analyse their equilibrium states; and consider actions of semigroups and groupoids. The project expects to produce significant mathematical outcomes, and the findings will be important beyond academia, expand Australia’s knowledge base and foster Australian competitiveness.Read moreRead less
States and structure of operator algebras from self-similar actions. Operator algebras are mathematical structures which have a wide range of applications including theoretical physics. This project will investigate a large class of interesting operator algebras associated to symmetries of graphs with fractal-like properties, detail their structure and classify the states associated to them.