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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Equations of Monge-Ampere type and applications. Many fundamental problems in geometry, physics and applied sciences are related to equations of Monge-Ampere type. In recent years there have been rapid developments in the study of these equations with major breakthroughs made by the proposers. This project aims at new discoveries and findings in theory and applications by resolving outstanding open problems, and enhance Australian leadership, expertise, and training in key areas of mathematics a ....Equations of Monge-Ampere type and applications. Many fundamental problems in geometry, physics and applied sciences are related to equations of Monge-Ampere type. In recent years there have been rapid developments in the study of these equations with major breakthroughs made by the proposers. This project aims at new discoveries and findings in theory and applications by resolving outstanding open problems, and enhance Australian leadership, expertise, and training in key areas of mathematics and its applications.Read moreRead less
Nonlinear elliptic partial differential equations and applications. Many fundamental advances in modern technology, science and economics are driven by the analysis of nonlinear models based on nonlinear partial differential equations. In recent years there has been increasing use in applications of partial differential equations of elliptic type with major discoveries made and longstanding problems resolved by the two Chief Investigators, who have in return received many international accolades ....Nonlinear elliptic partial differential equations and applications. Many fundamental advances in modern technology, science and economics are driven by the analysis of nonlinear models based on nonlinear partial differential equations. In recent years there has been increasing use in applications of partial differential equations of elliptic type with major discoveries made and longstanding problems resolved by the two Chief Investigators, who have in return received many international accolades. This project provides for the continuation of Australian leadership in key strategic areas of international science, such as optimal transportation, as well as the continued building of related expertise and training.Read moreRead less
Nonlinear elliptic equations and applications. Many fundamental advances in modern technology, science and economics are driven through the analysis of nonlinear models based on nonlinear partial differential equations. In recent years there has been an explosion in applications of partial differential equations of elliptic type with major discoveries in underlying theory being made by the two Chief Investigators. This project provides for the continuation of Australian leadership in key st ....Nonlinear elliptic equations and applications. Many fundamental advances in modern technology, science and economics are driven through the analysis of nonlinear models based on nonlinear partial differential equations. In recent years there has been an explosion in applications of partial differential equations of elliptic type with major discoveries in underlying theory being made by the two Chief Investigators. This project provides for the continuation of Australian leadership in key strategic areas of international science, such as optimal transportation, as well as the continued building of related expertise and training.Read moreRead less
Variational problems of Monge-Ampere type. Nonlinear models dominate the frontline of modern theoretical and applied mathematics. This project concerns contemporary variational problems with analysis linked strongly to the Monge-Ampere equation, which is a fully nonlinear partial differential equation. Its study in recent years has generated complex and deep theoretical issues along with a diverse range of applications. The proposal is divided into two themes, affine maximal surfaces (involving ....Variational problems of Monge-Ampere type. Nonlinear models dominate the frontline of modern theoretical and applied mathematics. This project concerns contemporary variational problems with analysis linked strongly to the Monge-Ampere equation, which is a fully nonlinear partial differential equation. Its study in recent years has generated complex and deep theoretical issues along with a diverse range of applications. The proposal is divided into two themes, affine maximal surfaces (involving fourth order partial differential equations of Monge-Ampere type) and optimal transportation (where Monge-Ampere theory has been applied successfully in recent years). Each of these builds upon major recent research breakthroughs of the proposers.Read moreRead less
Singular phenomena for nonlinear partial differential equations arising in applications. The development of nonlinear Partial Differential Equations (PDEs) in Australia is recognized worldwide through the outstanding contributions of mathematicians from the ANU, University of Sydney and other top Australian Universities. This project undertakes research in the PDEs field and follows directions of very current interest at an international level. Beyond the ANU, the project will enhance expertise ....Singular phenomena for nonlinear partial differential equations arising in applications. The development of nonlinear Partial Differential Equations (PDEs) in Australia is recognized worldwide through the outstanding contributions of mathematicians from the ANU, University of Sydney and other top Australian Universities. This project undertakes research in the PDEs field and follows directions of very current interest at an international level. Beyond the ANU, the project will enhance expertise in Australia in very active areas of mathematics research related to applications in physics, biology and other applied disciplines. Moreover, it will foster collaboration with mathematicians of international standing from Australia and abroad. Read moreRead less
Self-assembly and complexity: networks and patterns from materials to markets. Self-assembly leads the formation of patterns without external directing agents. It is responsible for the growth of complex multiscale structures found in biology and materials science and is a crucial concept for development of viable nanotechnologies. Complex systems, from biological ecosystems to financial markets and the Internet, are also characterized by spontaneous clustering and linkages that determine their ....Self-assembly and complexity: networks and patterns from materials to markets. Self-assembly leads the formation of patterns without external directing agents. It is responsible for the growth of complex multiscale structures found in biology and materials science and is a crucial concept for development of viable nanotechnologies. Complex systems, from biological ecosystems to financial markets and the Internet, are also characterized by spontaneous clustering and linkages that determine their collective behaviour. The project will investigate in detail the geometry, topology, materials science and statistical physics of networks, leading to design and characterization of robust self-assembled materials and complex systems.Read moreRead less
Analysis and applications of geometric evolution equations. This project will keep Australian research in geometric analysis at the leading edge of the field internationally. It will produce fundamental new insights in differential geometry and in the understanding of geometric partial differential equations, and will provide a rich and vigorous training ground for graduate and honours students.
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.