The Sakai scheme-Askey table correspondence, analogues of isomonodromy and determinantal point processes. The Australian mathematical sciences enjoys two research groups with active interests on Painleve equations in applied mathematics which are able to address difficult problems. Such a problem is to give a formulation of Sakai's 2001 classification of the Painleve equations in a form most suitable for applications. For this links will be made with a seemingly distinct area of mathematics - t ....The Sakai scheme-Askey table correspondence, analogues of isomonodromy and determinantal point processes. The Australian mathematical sciences enjoys two research groups with active interests on Painleve equations in applied mathematics which are able to address difficult problems. Such a problem is to give a formulation of Sakai's 2001 classification of the Painleve equations in a form most suitable for applications. For this links will be made with a seemingly distinct area of mathematics - the Askey table from the theory of hypergeometric orthogonal polynomials. A number of tractable PhD projects are suggested by this proposal.Read moreRead less
Real groups, Hodge theory, and the Langlands program. This mathematics project aims to settle open questions in real groups. The real groups are the fundamental symmetries occurring in nature and are important both in number theory and in the physical sciences. In particular, this project aims to reach a comprehensive understanding of Langlands duality for real groups, investigate how Hodge theory can be used to describe the unitary dual, and investigate the micro-local structure of systems of d ....Real groups, Hodge theory, and the Langlands program. This mathematics project aims to settle open questions in real groups. The real groups are the fundamental symmetries occurring in nature and are important both in number theory and in the physical sciences. In particular, this project aims to reach a comprehensive understanding of Langlands duality for real groups, investigate how Hodge theory can be used to describe the unitary dual, and investigate the micro-local structure of systems of differential equations. Potential benefits include increasing the international stature of mathematics in Australia and improving the quality of the workforce.Read moreRead less
Symmetries in real and complex geometry. This project concerns an important area of abstract modern geometry. The results and techniques of the project will lead to significant progress in this area. It will benefit the national scientific reputation, strengthen the research profile of the home institutions, and provide training to young researchers.