Representations of arithmetic groups and their associated zeta functions. This project aims to investigate deep connections between number theory and group theory by studying linear actions of arithmetic groups. Arithmetic groups are used in geometry, dynamics, number theory and other areas of pure mathematics. This project will study their representations from two perspectives. First, it will establish properties of the associated zeta functions to resolve open problems about the asymptotic beh ....Representations of arithmetic groups and their associated zeta functions. This project aims to investigate deep connections between number theory and group theory by studying linear actions of arithmetic groups. Arithmetic groups are used in geometry, dynamics, number theory and other areas of pure mathematics. This project will study their representations from two perspectives. First, it will establish properties of the associated zeta functions to resolve open problems about the asymptotic behaviour of the dimensions of the irreducible representations. Second, it will explore the evolution of representations across families of groups under new induction and restriction functors, in analogy with creation and annihilation operators in physics. The project will enhance Australia's capacity in representation theory and group theory, the mathematics that underline symmetry in nature.Read moreRead less
Braid groups and higher representation theory. Symmetry is a central notion in classical representation theory. In higher representation theory the symmetries of classical representation theory are replaced by higher symmetries. These higher symmetries contain new structure not present at the classical level. The proposed research will develop the higher representation theory of fundamental objects from classical representation theory and geometric group theory, focusing on braid groups and quan ....Braid groups and higher representation theory. Symmetry is a central notion in classical representation theory. In higher representation theory the symmetries of classical representation theory are replaced by higher symmetries. These higher symmetries contain new structure not present at the classical level. The proposed research will develop the higher representation theory of fundamental objects from classical representation theory and geometric group theory, focusing on braid groups and quantum groups.Read moreRead less
Conformal Differential Geometry. Differential geometry is a major branch of mathematics studying shape by using calculus and differential equations. This is a fundamental research project in this area, especially concerned with the interaction between geometry, differential equations, and symmetry. The mathematical notion of symmetry was already formalised early last century and nowadays lies at the very heart of mathematics and physics. Advances in this area provide essential tools in basic sci ....Conformal Differential Geometry. Differential geometry is a major branch of mathematics studying shape by using calculus and differential equations. This is a fundamental research project in this area, especially concerned with the interaction between geometry, differential equations, and symmetry. The mathematical notion of symmetry was already formalised early last century and nowadays lies at the very heart of mathematics and physics. Advances in this area provide essential tools in basic science and unexpected technological benefits can easily arise (for example, in medical imaging). Fundamental mathematical research is absolutely necessary if Australia is to maintain a presence on the international scientific stage.Read moreRead less
Indecomposable Structure in Representation Theory and Logarithmic Conformal Field Theory. Logarithmic conformal field theory describes non-local observables in statistical models of important physical systems (eg. polymers, percolation). This realisation has led to a recent explosion of activity among physicists and mathematicians. Mathematical physics in Australia is well-placed to capitalise on this activity, having several experts working in the area, and this project will significantly aug ....Indecomposable Structure in Representation Theory and Logarithmic Conformal Field Theory. Logarithmic conformal field theory describes non-local observables in statistical models of important physical systems (eg. polymers, percolation). This realisation has led to a recent explosion of activity among physicists and mathematicians. Mathematical physics in Australia is well-placed to capitalise on this activity, having several experts working in the area, and this project will significantly augment Australia's reputation within the international community by bringing (and developing) mathematical tools and insights which complement current research strengths. Such augmentations are vital to the well-being of mathematics and physics in Australia.Read moreRead less
Proper Group Actions in Complex Geometry. The results of the project will enhance Australia's performance in several key mathematical areas as well as mathematical applications to physics critical for expanding Australia's knowledge base and research capability. The project has strong international aspects, it will foster the international competitiveness of Australian research and establish long-term collaborations between Australian researchers and high profile world experts in the area of the ....Proper Group Actions in Complex Geometry. The results of the project will enhance Australia's performance in several key mathematical areas as well as mathematical applications to physics critical for expanding Australia's knowledge base and research capability. The project has strong international aspects, it will foster the international competitiveness of Australian research and establish long-term collaborations between Australian researchers and high profile world experts in the area of the proposal. It will create an opportunity for a Ph.D. graduate to be involved in top-class research as a Research Associate, and will attract Ph.D. and honours students thus enabling research training in a high-quality mathematical environment.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE120102369
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Higher representation theory. Representation theory lies at the very centre of mathematics, with applications in all areas of mathematics and mathematical physics; at some level it is about observing the symmetries of a system and exploiting them, and this has been invaluable. This project will explore the forefront of the modern, higher version of this research field.
Springer fibres, nilpotent cones and representation theory. This project will address new ideas and famous unsolved problems in the field of algebra known as representation theory, by studying the geometry of spaces called Springer fibres and nilpotent cones. This will keep Australian mathematics in the forefront of developments in this internationally active field, which is central to modern mathematics.
Geodetic groups: foundational problems in algebra and computer science. The project aims to resolve important and longstanding open problems in Geometric Group Theory and Theoretical Computer Science. Since the 1980s researchers have conjectured that the geometric property of being geodetic is equivalent to several purely algebraic, algorithmic, and language-theoretic characterisations.
The project team's expertise in geodesic properties of groups, the interaction between formal languages and g ....Geodetic groups: foundational problems in algebra and computer science. The project aims to resolve important and longstanding open problems in Geometric Group Theory and Theoretical Computer Science. Since the 1980s researchers have conjectured that the geometric property of being geodetic is equivalent to several purely algebraic, algorithmic, and language-theoretic characterisations.
The project team's expertise in geodesic properties of groups, the interaction between formal languages and groups, and the theory of rewriting systems, together with recent breakthroughs by the team ensures that significant results can be expected.
Benefits include training research students and postdoctoral researchers in cutting-edge techniques, and advancing fundamental knowledge in mathematics and computer science.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE140101825
Funder
Australian Research Council
Funding Amount
$334,710.00
Summary
The Algebraic Structure of Logarithmic Conformal Field Theory. Conformal field theory has given rise to a myriad of deep connections between physics and mathematics. Recently a generalisation of conformal field theory, called logarithmic conformal field theory, has garnered a lot of interest. These theories are necessary for understanding condensed matter systems with non-local observables such as percolation or polymers and for string theory on super group manifolds. This project will explore t ....The Algebraic Structure of Logarithmic Conformal Field Theory. Conformal field theory has given rise to a myriad of deep connections between physics and mathematics. Recently a generalisation of conformal field theory, called logarithmic conformal field theory, has garnered a lot of interest. These theories are necessary for understanding condensed matter systems with non-local observables such as percolation or polymers and for string theory on super group manifolds. This project will explore the algebraic structure of logarithmic conformal field theory. Expected outcomes include an improved understanding of how to systematically construct and solve logarithmic theories and will further consolidate Australia's reputation as an international centre for logarithmic conformal field theory.Read moreRead less