Topological Lambda-Algebras. This project will explore a mathematical puzzle that has defied twenty years of attempts to solve it. This puzzle is in an area which has been at the centre of worldwide mathematics research for more than a decade. The idea of the project is to use new techniques, developed in the last couple of years by Borger and Wieland, and bring them to bear on the old questions.
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
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
Dualities in String Theory and Conformal Field Theory in the context of the Geometric Langlands Program. The Langlands program ties together seemingly unrelated areas of Mathematics. Recently, in the context of the Geometric Langlands correspondence, novel connections with Theoretical Physics have emerged, thus becoming one of the most active areas of research in both Mathematics and Theoretical Physics. Australia has a number of world-renowned experts, including the two CI's, in various aspect ....Dualities in String Theory and Conformal Field Theory in the context of the Geometric Langlands Program. The Langlands program ties together seemingly unrelated areas of Mathematics. Recently, in the context of the Geometric Langlands correspondence, novel connections with Theoretical Physics have emerged, thus becoming one of the most active areas of research in both Mathematics and Theoretical Physics. Australia has a number of world-renowned experts, including the two CI's, in various aspects of the Langlands program, and is therefore well-placed to make seminal contributions. Being involved in these new developments is of crucial importance to the health of Mathematics and Theoretical Physics in Australia. An integral part of this proposal is student involvement and postgraduate training.Read moreRead less
From topological Hochschild homology to algebraic K-theory. The project will use methods of algebraic topology, specifically topological Hochschild homology, to study algebraic K-theory. This will increase our understanding of algebraic geometry, number theory, and the geometry of manifolds.
Modular Index Theory. This project capitilises on Australian advances in mathematics, particularly noncommutative geometry. It will maintain and extend Australia's prominence in this subject, providing excellent opportunities for young researchers via the research networks this project will establish. Being at the interface of ideas in mathematics and physics, there is potential for future technological spin offs for Australia.
Categorical symmetries in representation theory. This project aims to develop categorical symmetries of central objects in mathematics such as braid groups, the Hilbert scheme of points, and the Virasoro algebra. The concept of symmetry is an important organising principle in science. Representation theory is the field of mathematics concerned with studying symmetries. The problems proposed have connections to many different areas including algebra, geometry, topology, and mathematical physics. ....Categorical symmetries in representation theory. This project aims to develop categorical symmetries of central objects in mathematics such as braid groups, the Hilbert scheme of points, and the Virasoro algebra. The concept of symmetry is an important organising principle in science. Representation theory is the field of mathematics concerned with studying symmetries. The problems proposed have connections to many different areas including algebra, geometry, topology, and mathematical physics. This project expects to advance pure mathematics and provide potential benefit in many related fields.Read moreRead less
Algebraic K-theory and groups. This project will study the K-theory of division algebras, their generalisation to Azumaya algebras and the nonstable K-theory of rings. Expected outcomes would enhance our understanding on the structure of these K groups.
The goal is to settle some of the most significant conjectures in the subject: Bak's solvability of nonstable K groups over rings and the Merkurjev-Suslin conjectures on reduced K theory of division rings.
The study of these problems contribu ....Algebraic K-theory and groups. This project will study the K-theory of division algebras, their generalisation to Azumaya algebras and the nonstable K-theory of rings. Expected outcomes would enhance our understanding on the structure of these K groups.
The goal is to settle some of the most significant conjectures in the subject: Bak's solvability of nonstable K groups over rings and the Merkurjev-Suslin conjectures on reduced K theory of division rings.
The study of these problems contributes to and draws from such topics as group theory, commutative ring theory, algebraic number theory and algebraic geometry.
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