Global Behaviour of Integrable Complex Systems. Complex systems as diverse as the weather and the solar system are modelled by non-linear equations that have elusive, unstable solutions. An infinitesimally small change in the state of the system at one place can lead to a vast change in its behaviour far away. Such extreme sensitivity is often take to be a sign of chaos, but it also occurs in completely ordered, integrable systems. Our main aim is to tackle the immense challenge of describing th ....Global Behaviour of Integrable Complex Systems. Complex systems as diverse as the weather and the solar system are modelled by non-linear equations that have elusive, unstable solutions. An infinitesimally small change in the state of the system at one place can lead to a vast change in its behaviour far away. Such extreme sensitivity is often take to be a sign of chaos, but it also occurs in completely ordered, integrable systems. Our main aim is to tackle the immense challenge of describing the global behaviour of such elusive solutions, particularly when the systems depend on many variables.Read moreRead less
Iwasawa N Groups. Semisimple Lie groups and related objects are important in mathematics, theoretical physics (e.g., quantum mechanics and string theory), theoretical computer science (e.g., construction of expanders), and many other areas. They may be studied from different points of view---algebraic, analytic, geometric and representation theoretic---and these different studies find different applications. The project aims to synthesize the different points of view, to understand their funda ....Iwasawa N Groups. Semisimple Lie groups and related objects are important in mathematics, theoretical physics (e.g., quantum mechanics and string theory), theoretical computer science (e.g., construction of expanders), and many other areas. They may be studied from different points of view---algebraic, analytic, geometric and representation theoretic---and these different studies find different applications. The project aims to synthesize the different points of view, to understand their fundamental unity, and to allow results of one type to be translated into another context.Read moreRead less
On the Geometry of Liquid Crystals and Biological Membranes. This project will provide fundamental insights via realistic mathematical models into two areas of technological importance in the development of certain advanced materials involving liquid crystals and biomembranes. The use of liquid crystal devices is ubiquitous in the design of optical display units. Biomembranes are of much current importance, in particular, in connection with sophisticated drug delivery systems. The design of adva ....On the Geometry of Liquid Crystals and Biological Membranes. This project will provide fundamental insights via realistic mathematical models into two areas of technological importance in the development of certain advanced materials involving liquid crystals and biomembranes. The use of liquid crystal devices is ubiquitous in the design of optical display units. Biomembranes are of much current importance, in particular, in connection with sophisticated drug delivery systems. The design of advanced `smart' materials which admit solitonic behaviour is an area at the forefront of materials science and as such is important to the continued development of an advanced technological base within Australia.Read moreRead less
Noncommutative geometry in representation theory and quantum physics. One of the most important problems in natural science is to understand the structure of spacetime at the Planck scale. Mathematical investigations in recent years have predicted that at this scale, spacetime becomes noncommutative. Taking this noncommutativity into account, the project brings together geometry, algebra and quantum mechanics to develop new mathematical theories required for addressing the problem. It promises ....Noncommutative geometry in representation theory and quantum physics. One of the most important problems in natural science is to understand the structure of spacetime at the Planck scale. Mathematical investigations in recent years have predicted that at this scale, spacetime becomes noncommutative. Taking this noncommutativity into account, the project brings together geometry, algebra and quantum mechanics to develop new mathematical theories required for addressing the problem. It promises to make fundamental contributions to both mathematics and theoretical physics. Read moreRead less
Quantum vertex algebras. The project aims to address major mathematical problems on the structure and representations of the families of quantum groups and vertex algebras associated with Lie algebras. Originating from solvable lattice models in statistical mechanics, the theory of quantum groups has important connections with, and applications to, a wide range of subjects in mathematics and physics. The project will extend and develop explicit theory of both the classical and quantum versions o ....Quantum vertex algebras. The project aims to address major mathematical problems on the structure and representations of the families of quantum groups and vertex algebras associated with Lie algebras. Originating from solvable lattice models in statistical mechanics, the theory of quantum groups has important connections with, and applications to, a wide range of subjects in mathematics and physics. The project will extend and develop explicit theory of both the classical and quantum versions of the vertex algebras which are of importance to conformal field theory and soliton spin-chain models.Read moreRead less
Extremal problems in hypergraph matchings. Matchings in hypergraphs are a way of understanding complex relationships between objects in any set. This project will develop a mathematical theory that covers both extreme and typical cases. This theory will have applications wherever hypergraphs are used as models, for example in machine learning, game theory, databases, data mining and optimisation.
Enumeration and properties of large discrete structures. This project aims to study a fundamental property of random graphs, by further developing a recently introduced approach to the problem of enumerating graphs with given degrees. Using this new method, the project expects to generate new knowledge on the number of connections that each node has with other nodes in a random graph, and to develop new strategies for counting the graphs or networks with a given property. The project expects to ....Enumeration and properties of large discrete structures. This project aims to study a fundamental property of random graphs, by further developing a recently introduced approach to the problem of enumerating graphs with given degrees. Using this new method, the project expects to generate new knowledge on the number of connections that each node has with other nodes in a random graph, and to develop new strategies for counting the graphs or networks with a given property. The project expects to produce new theoretical results as well as enhanced capabilities of mathematical research. Potential benefits arise through the uses of these theoretical combinatorial objects to study naturally occurring networks such as social networks, the network of the world wide web, and chemical compounds.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE170100789
Funder
Australian Research Council
Funding Amount
$324,499.00
Summary
Advances in graph Ramsey theory. This project aims to solve significant questions at the forefront of graph Ramsey theory, which provides the theoretical background for understanding networks that are omnipresent in the modern world. Major progress is anticipated on the recently introduced concept of Ramsey equivalence, including the development of deep new tools that combine probabilistic methods, extremal graph theory and graph decomposition techniques. The project will use these new tools to ....Advances in graph Ramsey theory. This project aims to solve significant questions at the forefront of graph Ramsey theory, which provides the theoretical background for understanding networks that are omnipresent in the modern world. Major progress is anticipated on the recently introduced concept of Ramsey equivalence, including the development of deep new tools that combine probabilistic methods, extremal graph theory and graph decomposition techniques. The project will use these new tools to solve old questions on the structure of minimal Ramsey graphs, thus fostering the international competitiveness of Australian research and enhancing Australia's reputation as a knowledge nation.Read moreRead less
Enumeration and random generation of contingency tables with given margins. This project aims to find algorithms to construct random tables of numbers having given totals across the rows and down the columns. The aim is also to study properties of such tables. A significant aspect of the project is that it is expected to cover scenarios where all existing methods fail, by deploying recently developed powerful techniques used for random networks in combinatorics. Expected outcomes of this project ....Enumeration and random generation of contingency tables with given margins. This project aims to find algorithms to construct random tables of numbers having given totals across the rows and down the columns. The aim is also to study properties of such tables. A significant aspect of the project is that it is expected to cover scenarios where all existing methods fail, by deploying recently developed powerful techniques used for random networks in combinatorics. Expected outcomes of this project include the development of efficient algorithms that can be used in statistics for identifying relationships between variables in large data sets. This would help bring Australia to the forefront of research in an area that is significant both in data analysis and in discrete mathematics.
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Discovery Early Career Researcher Award - Grant ID: DE150100030
Funder
Australian Research Council
Funding Amount
$300,000.00
Summary
A new concept of independence in noncommutative probability theory. The concept of independence lies at the very core of the probability theory. Many attempts to establish the general notion of independence in noncommutative probability theory have led to only two examples so far: the classical (commutative) independence and the free one introduced by Voiculescu. Every other approach has failed to demonstrate the analogues of the key probabilistic results, such as the Law of Large Numbers and th ....A new concept of independence in noncommutative probability theory. The concept of independence lies at the very core of the probability theory. Many attempts to establish the general notion of independence in noncommutative probability theory have led to only two examples so far: the classical (commutative) independence and the free one introduced by Voiculescu. Every other approach has failed to demonstrate the analogues of the key probabilistic results, such as the Law of Large Numbers and the Central Limit Theorem. There is an urgent need for new efficient methodology. This project aims to develop an approach to the independence in terms of mixed momenta and to find new examples of independence besides the ones mentioned above.Read moreRead less