Monge-Ampere equations and applications. The Monge-Ampere equation is a premier fully nonlinear partial differential equation with significant applications in geometry, physics and applied science. Building upon breakthroughs made by the proposers in previous grant research, this project aims to resolve challenging problems involving Monge-Ampere type equations and applications. The project goal is to establish new regularity theory and classify singularity profile for solutions to Monge-Ampere ....Monge-Ampere equations and applications. The Monge-Ampere equation is a premier fully nonlinear partial differential equation with significant applications in geometry, physics and applied science. Building upon breakthroughs made by the proposers in previous grant research, this project aims to resolve challenging problems involving Monge-Ampere type equations and applications. The project goal is to establish new regularity theory and classify singularity profile for solutions to Monge-Ampere type equation arising in applied sciences, by introducing new ideas and developing innovative cutting-edge techniques. Expected outcomes include resolution of outstanding open problems and continuing enhancement of Australian leadership and expertise in a major area of mathematics.
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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
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.
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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Discovery Early Career Researcher Award - Grant ID: DE120102369
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Higher representation theory. Representation theory lies at the very centre of mathematics, with applications in all areas of mathematics and mathematical physics; at some level it is about observing the symmetries of a system and exploiting them, and this has been invaluable. This project will explore the forefront of the modern, higher version of this research field.
Discovery Early Career Researcher Award - Grant ID: DE140101825
Funder
Australian Research Council
Funding Amount
$334,710.00
Summary
The Algebraic Structure of Logarithmic Conformal Field Theory. Conformal field theory has given rise to a myriad of deep connections between physics and mathematics. Recently a generalisation of conformal field theory, called logarithmic conformal field theory, has garnered a lot of interest. These theories are necessary for understanding condensed matter systems with non-local observables such as percolation or polymers and for string theory on super group manifolds. This project will explore t ....The Algebraic Structure of Logarithmic Conformal Field Theory. Conformal field theory has given rise to a myriad of deep connections between physics and mathematics. Recently a generalisation of conformal field theory, called logarithmic conformal field theory, has garnered a lot of interest. These theories are necessary for understanding condensed matter systems with non-local observables such as percolation or polymers and for string theory on super group manifolds. This project will explore the algebraic structure of logarithmic conformal field theory. Expected outcomes include an improved understanding of how to systematically construct and solve logarithmic theories and will further consolidate Australia's reputation as an international centre for logarithmic conformal field theory.Read moreRead less