Monge-Ampere type equations and their applications. The study of Monge-Ampere equations has attracted major attention in mathematics in recent years, due to many significant applications in geometry, physics and applied science. This project aims to resolve challenging problems involving Monge-Ampere type equations, by utilising new ideas and breakthroughs made by the proposer. A comprehensive regularity theory for Monge-Ampere type equations, particularly in the degenerate case, is expected to ....Monge-Ampere type equations and their applications. The study of Monge-Ampere equations has attracted major attention in mathematics in recent years, due to many significant applications in geometry, physics and applied science. This project aims to resolve challenging problems involving Monge-Ampere type equations, by utilising new ideas and breakthroughs made by the proposer. A comprehensive regularity theory for Monge-Ampere type equations, particularly in the degenerate case, is expected to be established. Innovative cutting-edge techniques and interdisciplinary approaches are expected to be developed. Anticipated outcomes of this project include the resolution of outstanding open problems and continuing enhancement of Australian leadership and expertise in a major area of mathematics.Read moreRead less
Singularity and regularity for Monge-Ampere type equations. The Monge-Ampere equation, as a premier nonlinear partial differential equation, arises in several areas including geometry, physics, and optimal transportation. Many important problems and applications are related to the regularity of solutions, which are obstructed by singularities. This project aims to classify the geometry of the singular sets, and to establish a comprehensive regularity theory for general Monge-Ampere type equation ....Singularity and regularity for Monge-Ampere type equations. The Monge-Ampere equation, as a premier nonlinear partial differential equation, arises in several areas including geometry, physics, and optimal transportation. Many important problems and applications are related to the regularity of solutions, which are obstructed by singularities. This project aims to classify the geometry of the singular sets, and to establish a comprehensive regularity theory for general Monge-Ampere type equations by using innovative approaches and developing cutting-edge technologies in partial differential equations. Expected outcomes include the resolution of outstanding open problems. This project will significantly enhance Australia’s leadership and expertise in a major area of mathematics and applications.Read moreRead less
Deformation of singularities through Hodge theory and derived categories. Moduli theory, the modern classification theory of mathematical objects, is a branch of algebraic geometry with applications in wide-ranging areas from the theoretical high-energy physics (dark matter and Higgs boson) to data encryption and correction via cryptography. The aim of this project is to resolve central open problems in this theory. This will be achieved by developing new methods and establishing deeper connecti ....Deformation of singularities through Hodge theory and derived categories. Moduli theory, the modern classification theory of mathematical objects, is a branch of algebraic geometry with applications in wide-ranging areas from the theoretical high-energy physics (dark matter and Higgs boson) to data encryption and correction via cryptography. The aim of this project is to resolve central open problems in this theory. This will be achieved by developing new methods and establishing deeper connections between various dynamic branches of these fields. By undertaking research at the forefronts of these highly active areas, this project will both strengthen the current expertise within the Australian mathematical community and precipitate the advance of Australian high-tech industries. Read moreRead less
Interplay between Ergodic Theory, Additive Combinatorics and Ramsey Theory. This project aims to address fundamental problems in Number Theory and Combinatorics by developing new innovative ergodic theoretic methods. Expected outcomes of the project include finding new patterns in dense subsets of trees, obtaining rigorous number-theoretic results emphasising the independence of addition and multiplication, finding infinite patterns in dense subsets of primes, and developing a multi-dimensional ....Interplay between Ergodic Theory, Additive Combinatorics and Ramsey Theory. This project aims to address fundamental problems in Number Theory and Combinatorics by developing new innovative ergodic theoretic methods. Expected outcomes of the project include finding new patterns in dense subsets of trees, obtaining rigorous number-theoretic results emphasising the independence of addition and multiplication, finding infinite patterns in dense subsets of primes, and developing a multi-dimensional analogue of the dense model theory for primes. This project will provide significant benefits to Australian research via an intensive collaboration with best international and Australian researchers working in ergodic and number theory as well as will be used to educate a new generation of Australian students. Read moreRead less
Categorical geometry and perfect group schemes. The aims of this project are to construct novel geometric theories based on newly discovered tensor categories, to apply the theories to solve open problems in representation theory, algebra and category theory, and to establish profitable new connections between the influential theories of affine group schemes and classifying spaces. The geometric theories will be developed in a universal way, generalising both classical algebraic geometry and sup ....Categorical geometry and perfect group schemes. The aims of this project are to construct novel geometric theories based on newly discovered tensor categories, to apply the theories to solve open problems in representation theory, algebra and category theory, and to establish profitable new connections between the influential theories of affine group schemes and classifying spaces. The geometric theories will be developed in a universal way, generalising both classical algebraic geometry and super geometry from physics, and specialising to infinitely many new theories. This universality ensures a significantly broader basis for long term applications of geometry in many areas of science. Other benefits include enhanced international collaboration and scientific capacity in Australia.Read moreRead less
A Functional Analysis of the Hypoelliptic Laplacian. Strike a bell, a sphere, or any geometrical object, and it rings. The frequencies of ringing are the mathematical spectrum, which encodes deep secrets about the shape of the object. The spectrum of the hypoelliptic laplacian is known to carry deep truths in mathematics and physics, but it remains difficult to understand. We propose a new analytic foundation, which will replace the so far non-analytical ad hoc approach, and make accessible many ....A Functional Analysis of the Hypoelliptic Laplacian. Strike a bell, a sphere, or any geometrical object, and it rings. The frequencies of ringing are the mathematical spectrum, which encodes deep secrets about the shape of the object. The spectrum of the hypoelliptic laplacian is known to carry deep truths in mathematics and physics, but it remains difficult to understand. We propose a new analytic foundation, which will replace the so far non-analytical ad hoc approach, and make accessible many new results. It is key to better understanding differential equations which lie at the boundary between quantum mechanics and the classical world. This will pave the way for Australian leadership in a new century of differential equations and geometry, and training of young mathematicians.Read moreRead less
Mathematics for future magnetic devices. The aim of this project is to develop a mathematical theory and numerical models of stochastic partial differential
equations for magnetic nano-structures. Such materials will yield next-generation magnetic memories with up to
three orders of magnitude faster switching speeds and dramatically increased data storage density. New
mathematical theories will help understand their sensitivity to small random fluctuations that can destroy stored
information. Th ....Mathematics for future magnetic devices. The aim of this project is to develop a mathematical theory and numerical models of stochastic partial differential
equations for magnetic nano-structures. Such materials will yield next-generation magnetic memories with up to
three orders of magnitude faster switching speeds and dramatically increased data storage density. New
mathematical theories will help understand their sensitivity to small random fluctuations that can destroy stored
information. This project aims to revolutionise mathematical modelling of magnetic memories and put Australia at
the forefront of international research. Technological advances to create much smaller and faster memory devices
are expected to enable groundbreaking ways of managing and mining big dataRead moreRead less