Representation theory of groups and applications to geometry and number theory. Representation theory is at the center of the mathematical study of symmetry, which we constantly use to understand the world. Combine with geometry this theory produces spectacular results in number theory. This project aims to study p-adic phenomena in these theories. Its
main outcomes will be p-adic automorphic forms and local functoriality.
Categorical splitting theorems in algebraic geometry. Algebraic geometry is the study of solutions of polynomial equations.
It is one of the richest fields of Mathematics, and has led to advances in cryptography and other areas of technology. The project aims to apply in this field some recently developed abstract techniques in order to obtain results of a new type. It is also expected that the approach taken will help to simplify and unify the
branches of algebraic geometry considered. Proje ....Categorical splitting theorems in algebraic geometry. Algebraic geometry is the study of solutions of polynomial equations.
It is one of the richest fields of Mathematics, and has led to advances in cryptography and other areas of technology. The project aims to apply in this field some recently developed abstract techniques in order to obtain results of a new type. It is also expected that the approach taken will help to simplify and unify the
branches of algebraic geometry considered. Projects of a theoretical
nature such as this one help to maintain Australia's high standing in
the international scientific community.Read moreRead less
Noncommutative geometry of groups acting on buildings. Consider a tiling of the plane by triangles, where each triangle is labeled by an element of a finite alphabet. Suppose that only certain pairs of labels are allowed to be adjacent to each other in each direction. The tiled planes can be pasted together to form the abstract mathematical object known as a building. This building and its boundary, give rise to new families of C*-algebras and groups. The algebras have a rich structure which it ....Noncommutative geometry of groups acting on buildings. Consider a tiling of the plane by triangles, where each triangle is labeled by an element of a finite alphabet. Suppose that only certain pairs of labels are allowed to be adjacent to each other in each direction. The tiled planes can be pasted together to form the abstract mathematical object known as a building. This building and its boundary, give rise to new families of C*-algebras and groups. The algebras have a rich structure which it is proposed to investigate and link with geometric properties of the groups. New insights into geometry, dynamics and algebra are expected.Read moreRead less
Higher Line Bundles in Geometry and Physics. This project seeks to develop a theory of geometric objects, `higher line bundles', which realise elements of higher dimensional cohomology groups. In particular this project will develop a theory of differential geometry for these objects, allowing one to interpret differential forms representing cohomology classes as the `curvature' of a higher line bundle. This will have applications in quantum field theory and string/brane theory.
Triangulated categories and their applications. This project is at the cutting edge of modern, international research in mathematics. Having work of this calibre done in Australia raises our international prestige, and makes Australia a more attractive place for top-notch hi-tech companies. Furthermore, training our young people to such a high standard will have the long-term effect of raising our profile.
Derived categories and their applications, especially in K-theory, topology and algebraic geometry. Algebraic geometry, topology and algebraic K-theory are mathematical disciplines that study different aspects of geometry. In all these areas of study, derived categories have proved to be powerful tools. This project aims to use derived categories to advance our understanding of geometry. Involved are some of the main open questions in geometry from the second half of the twentieth century.
....Derived categories and their applications, especially in K-theory, topology and algebraic geometry. Algebraic geometry, topology and algebraic K-theory are mathematical disciplines that study different aspects of geometry. In all these areas of study, derived categories have proved to be powerful tools. This project aims to use derived categories to advance our understanding of geometry. Involved are some of the main open questions in geometry from the second half of the twentieth century.
The research is being nominated for the Complex/Intelligent Systems Priority Area. Geometry is relevant in two ways. Secure and/or error correcting codes are often based on algebraic geometry. And modelling concurrency problems involves homotopy theory.Read moreRead less
Global aspects of dualities in String Theory in the presence of background fluxes. String Theory, known to the general public as the "Theory of Everything', is currently an extremely active area of research internationally. It has not only stimulated considerable interaction between mathematical physicists and mathematicians, but also increased public interest in science through television programs and books. Unfortunately, the majority of the Australian scientific community has not yet caught ....Global aspects of dualities in String Theory in the presence of background fluxes. String Theory, known to the general public as the "Theory of Everything', is currently an extremely active area of research internationally. It has not only stimulated considerable interaction between mathematical physicists and mathematicians, but also increased public interest in science through television programs and books. Unfortunately, the majority of the Australian scientific community has not yet caught up with these developments. Our recent papers, all published in premier journals in this field, have not only received widespread international attention but have also increased the profile of String Theory amongst Australia's mathematicians and mathematical physicists. The proposed project is expected to continue this trend.Read moreRead less
Stability conditions on triangulated categories and related aspects of homological mirror symmetry. The proposed research studies one of the deepest questions in nature through superstring theory and mathematics with leading experts around the world. So, the proposed project maintains the Australia's profile in science. Also, the proposed project fits within the the Research Priority: Frontier Technologies for Building and Transforming Australian Industries. We will have exciting mathematical di ....Stability conditions on triangulated categories and related aspects of homological mirror symmetry. The proposed research studies one of the deepest questions in nature through superstring theory and mathematics with leading experts around the world. So, the proposed project maintains the Australia's profile in science. Also, the proposed project fits within the the Research Priority: Frontier Technologies for Building and Transforming Australian Industries. We will have exciting mathematical discussions which stimulate Australian students. They will be able to take advantage of such experience, especially when they need innovation. Thus, it is an investment for future of Australian industries.Read moreRead less
Generalized Geometries and their Applications. Geometry is one of the pillars of both ancient and modern mathematics. It also plays a vital role in many scientific applications, in particular in physics. Progress on the mathematical aspects and the applications have often gone hand in hand, as for example with differential geometry and general relativity. Geometry is a very fruitful area for interdisciplinary research.
Australia has a long tradition and a recognized research strength in Mat ....Generalized Geometries and their Applications. Geometry is one of the pillars of both ancient and modern mathematics. It also plays a vital role in many scientific applications, in particular in physics. Progress on the mathematical aspects and the applications have often gone hand in hand, as for example with differential geometry and general relativity. Geometry is a very fruitful area for interdisciplinary research.
Australia has a long tradition and a recognized research strength in Mathematical Physics, and this project will contribute to maintaining that status. An integral part of this proposal is student involvement and postgraduate research training, for which the topic lends itself particularly well.Read moreRead less
Twisted K-theory and its application to String Theory and Conformal Field Theory. String Theory is, at present, the only consistent theory of quantum gravity. Recently, twisted K-theory was proposed as the algebraic structure underlying the classification of D-branes, i.e. solitonic extended objects, in certain closed string backgrounds. In this project we aim to advance our understanding of the properties of twisted K-theory in the context of String Theory and Conformal Field Theory. The ult ....Twisted K-theory and its application to String Theory and Conformal Field Theory. String Theory is, at present, the only consistent theory of quantum gravity. Recently, twisted K-theory was proposed as the algebraic structure underlying the classification of D-branes, i.e. solitonic extended objects, in certain closed string backgrounds. In this project we aim to advance our understanding of the properties of twisted K-theory in the context of String Theory and Conformal Field Theory. The ultimate goal is to find the appropriate K-theory classifying D-branes in arbitrary closed string backgrounds or, similarly, classifying boundary Conformal Field Theories. It has already emerged that the K-theory of C*-algebras will play an important role.Read moreRead less