Minimal surfaces. Recent stunning progress in topology, in particular a possible solution to one of the Clay Institute million dollar problems, using techniques from partial differential equations and minimal surfaces has made this area a hot topic. To attract researchers in this field to visit Australia and to train students in this area is a major part of this project.
Global Behaviour of Integrable Complex Systems. Complex systems as diverse as the weather and the solar system are modelled by non-linear equations that have elusive, unstable solutions. An infinitesimally small change in the state of the system at one place can lead to a vast change in its behaviour far away. Such extreme sensitivity is often take to be a sign of chaos, but it also occurs in completely ordered, integrable systems. Our main aim is to tackle the immense challenge of describing th ....Global Behaviour of Integrable Complex Systems. Complex systems as diverse as the weather and the solar system are modelled by non-linear equations that have elusive, unstable solutions. An infinitesimally small change in the state of the system at one place can lead to a vast change in its behaviour far away. Such extreme sensitivity is often take to be a sign of chaos, but it also occurs in completely ordered, integrable systems. Our main aim is to tackle the immense challenge of describing the global behaviour of such elusive solutions, particularly when the systems depend on many variables.Read moreRead less
Iwasawa N Groups. Semisimple Lie groups and related objects are important in mathematics, theoretical physics (e.g., quantum mechanics and string theory), theoretical computer science (e.g., construction of expanders), and many other areas. They may be studied from different points of view---algebraic, analytic, geometric and representation theoretic---and these different studies find different applications. The project aims to synthesize the different points of view, to understand their funda ....Iwasawa N Groups. Semisimple Lie groups and related objects are important in mathematics, theoretical physics (e.g., quantum mechanics and string theory), theoretical computer science (e.g., construction of expanders), and many other areas. They may be studied from different points of view---algebraic, analytic, geometric and representation theoretic---and these different studies find different applications. The project aims to synthesize the different points of view, to understand their fundamental unity, and to allow results of one type to be translated into another context.Read moreRead less
Equivalence Relations, Group Actions, and Descriptive Set Theory. This project is a contribution to basic and foundational research in the area of Pure Mathematics generally and Mathematical Logic specifically. Logic in particular appears in disciplines as diverse as Computer Science,Linguistics, and Philosophy, and the development of logic in these fields has been profoundly influenced by the foundational work of mathematical logicians. The innovative techniques introduced in this proposal will ....Equivalence Relations, Group Actions, and Descriptive Set Theory. This project is a contribution to basic and foundational research in the area of Pure Mathematics generally and Mathematical Logic specifically. Logic in particular appears in disciplines as diverse as Computer Science,Linguistics, and Philosophy, and the development of logic in these fields has been profoundly influenced by the foundational work of mathematical logicians. The innovative techniques introduced in this proposal will enable Australia to maintain a position at the forefront of Pure Mathematics, and by recruiting a recent winner of the highly prestigious Karp prize the country will be instantly established as one of the leading centers of Mathematical Logic.Read moreRead less
Geometric structures in representation theory. Mathematics underpins every aspect of people's interactions with nature (e.g. physics) and with each other (e.g. finance). Its uses range from formulating physical laws in order to understand and predict nature, to analysis of financial concepts and transactions. This project will formulate and develop three new fundamental mathematical concepts: cellular algebras, eigenspace geometries, and diagram algebras. Benefits include enhancement of Australi ....Geometric structures in representation theory. Mathematics underpins every aspect of people's interactions with nature (e.g. physics) and with each other (e.g. finance). Its uses range from formulating physical laws in order to understand and predict nature, to analysis of financial concepts and transactions. This project will formulate and develop three new fundamental mathematical concepts: cellular algebras, eigenspace geometries, and diagram algebras. Benefits include enhancement of Australia's position at the very frontier of world class mathematical research, and a myriad of potential applications to physics, coding theory, information technology, electronic security and experimental design.Read moreRead less
The geometry of exotic nilpotent cones. This research will describe the geometry of some important objects which sit at the boundary of algebra, geometry, and combinatorics. It has intrinsic value as a significant addition to the heritage of mathematical thought, and will strengthen Australian traditions in these areas of mathematics.
Pyramids and decomposition numbers for the symmetric and general linear groups. This project takes a novel approach to the decomposition number problem for the symmetric and general linear groups by setting up a new framework for computing them using the combinatorics of pyramids. The decomposition numbers of an algebra are an important statistic which gives detailed structural information about its representations. These numbers can be used to compute the dimensions of the irreducible represen ....Pyramids and decomposition numbers for the symmetric and general linear groups. This project takes a novel approach to the decomposition number problem for the symmetric and general linear groups by setting up a new framework for computing them using the combinatorics of pyramids. The decomposition numbers of an algebra are an important statistic which gives detailed structural information about its representations. These numbers can be used to compute the dimensions of the irreducible representations of the algebra and they play an important role in the applications of representation theory to other fields such as knot theory and statistical mechanics.Read moreRead less
Algebras with Frobenius morphisms and quantum groups. In this digitalized world, our life relies on mathematics more than ever. Counting and numbers are just one example of this. Another is the public key codes for online payments and transactions. Mathematics is of enormous importance in this technology dominated age. This proposal is to carry out high level mathematical research in Australia. Basic research on quantum groups underpins applied research and certain areas such as quantum mechanic ....Algebras with Frobenius morphisms and quantum groups. In this digitalized world, our life relies on mathematics more than ever. Counting and numbers are just one example of this. Another is the public key codes for online payments and transactions. Mathematics is of enormous importance in this technology dominated age. This proposal is to carry out high level mathematical research in Australia. Basic research on quantum groups underpins applied research and certain areas such as quantum mechanics and string theory. Some structure of quantum groups is too complicated to be seen by even a professional mathematician. A possible interpretation by using representations over a finite field would make it more usable and accessible by computer.Read moreRead less
New Geometric and Entropy Techniques for Differential Equations. The three main practical outcomes of this mathematical research will be better predictability of salt movement responsible for land degradation, better predictability of surface evolution of microelectronic components in nanoscale technology and an open source computer package that harnesses new and powerful geometrical techniques to solve differential equations. The project will train the next generation of researchers in the math ....New Geometric and Entropy Techniques for Differential Equations. The three main practical outcomes of this mathematical research will be better predictability of salt movement responsible for land degradation, better predictability of surface evolution of microelectronic components in nanoscale technology and an open source computer package that harnesses new and powerful geometrical techniques to solve differential equations. The project will train the next generation of researchers in the mathematical modelling of critical physical processes and it will bring international experts to Australia to work on these vital problems.Read moreRead less
Modular representations of cyclotomic algebras. This project addresses cutting edge questions in the representation theory of cyclotomic Hecke algebras. Our main focus will be computing decomposition matrices for these algebras. We approach this question from several different directions, each of which will give new insights and lead to significant advances in the theory. The decomposition number problem is important because its' solution gives deep structural information about these algebras wh ....Modular representations of cyclotomic algebras. This project addresses cutting edge questions in the representation theory of cyclotomic Hecke algebras. Our main focus will be computing decomposition matrices for these algebras. We approach this question from several different directions, each of which will give new insights and lead to significant advances in the theory. The decomposition number problem is important because its' solution gives deep structural information about these algebras which can then be applied in other areas. This project will have high impact because cyclotomic Hecke algebras have applications in many different areas and they are currently a hot topic of research in mathematics.Read moreRead less