New Geometric and Entropy Techniques for Differential Equations. The three main practical outcomes of this mathematical research will be better predictability of salt movement responsible for land degradation, better predictability of surface evolution of microelectronic components in nanoscale technology and an open source computer package that harnesses new and powerful geometrical techniques to solve differential equations. The project will train the next generation of researchers in the math ....New Geometric and Entropy Techniques for Differential Equations. The three main practical outcomes of this mathematical research will be better predictability of salt movement responsible for land degradation, better predictability of surface evolution of microelectronic components in nanoscale technology and an open source computer package that harnesses new and powerful geometrical techniques to solve differential equations. The project will train the next generation of researchers in the mathematical modelling of critical physical processes and it will bring international experts to Australia to work on these vital problems.Read moreRead less
Representation Theory: Path models and decompositions. The research in this proposal develops tools for capitalising on the benefits of symmetry in large complex systems. These techniques and processes are applicable for solving complex problems in large interactive systems. This project will involve young researchers and train them for problem solving in a wealth of fields, including management, the sciences, the financial industries, and the development of technologies. The research is in o ....Representation Theory: Path models and decompositions. The research in this proposal develops tools for capitalising on the benefits of symmetry in large complex systems. These techniques and processes are applicable for solving complex problems in large interactive systems. This project will involve young researchers and train them for problem solving in a wealth of fields, including management, the sciences, the financial industries, and the development of technologies. The research is in one of the most active cutting edge areas of pure mathematics and will contribute to maintaining Australia's position as a leading nationality in research in representation theory and its applications.
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Geometric Group Theory. Groups arise naturally as symmetries of geometric objects. Often groups have an interesting geometric structure obtained by thinking of these geometric objects coursely. This project aims to study the subgroup structure of such groups and obtain homological, geometric and algorithmic information. It further investigates natural decompositions of groups with geometric structure along special subgroups so that the factors have simpler properties.{P
Rigidity in measured group theory and geometric group theory. Elite universities throughout the world have all made a point of being leaders in the field of pure mathematics. Geometric group theory and orbit equivalence are currently topical areas which attract many of the best young pure mathematicians as is demonstrated by recent invited talks at the International Congress of Mathematicians. This project will foster the development of these fields in Australia as well as nurturing existing e ....Rigidity in measured group theory and geometric group theory. Elite universities throughout the world have all made a point of being leaders in the field of pure mathematics. Geometric group theory and orbit equivalence are currently topical areas which attract many of the best young pure mathematicians as is demonstrated by recent invited talks at the International Congress of Mathematicians. This project will foster the development of these fields in Australia as well as nurturing existing efforts and international links. This proposal will also provide training and research experience for Australian honours and graduate students in mathematics.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE160100975
Funder
Australian Research Council
Funding Amount
$307,536.00
Summary
Algebraic groups and Springer theory. This project aims to explore representation theory, which is a study of the basic symmetries that occur in nature. By its nature, representation theory has applications to number theory, physics, national security and internet security, and other sciences. Generalised Springer theory plays an important role in representations of finite groups of Lie type. The project aims to develop an analogous theory in a more general setting that includes symmetric spaces ....Algebraic groups and Springer theory. This project aims to explore representation theory, which is a study of the basic symmetries that occur in nature. By its nature, representation theory has applications to number theory, physics, national security and internet security, and other sciences. Generalised Springer theory plays an important role in representations of finite groups of Lie type. The project aims to develop an analogous theory in a more general setting that includes symmetric spaces. Moreover, the project aims to address various outstanding problems in algebraic groups. The project also plans to explore the connection between the geometry of certain null-cones and deformations of Galois representations.Read moreRead less
Macdonald polynomials: Combinatorics and representations. This proposal is part of the aim to build a world class research team in algebraic combinatorics and combinatorial representation theory at the University of Melbourne, led by the two CI.
These fields are currently experiencing very rapid growth and development, and a strong Australia based team will further enhance the country's strong reputation in combinatorics and algebra.
The project will also provide a perfect training ground fo ....Macdonald polynomials: Combinatorics and representations. This proposal is part of the aim to build a world class research team in algebraic combinatorics and combinatorial representation theory at the University of Melbourne, led by the two CI.
These fields are currently experiencing very rapid growth and development, and a strong Australia based team will further enhance the country's strong reputation in combinatorics and algebra.
The project will also provide a perfect training ground for Higher Degree Students with interests in pure mathematics as well as computer
algebra and symbolic computation.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE190101231
Funder
Australian Research Council
Funding Amount
$390,000.00
Summary
Integrable systems from gauge theories. This project aims to construct and describe three important integrable systems and to produce new significant results in representation theory and mathematical physics. The focus of the project is the theory of integrable systems, a breakthrough area in modern mathematics, and it will use tools from geometric representation theory to study the interrelation between geometry and mathematical physics. Originating from string theory, the project will not only ....Integrable systems from gauge theories. This project aims to construct and describe three important integrable systems and to produce new significant results in representation theory and mathematical physics. The focus of the project is the theory of integrable systems, a breakthrough area in modern mathematics, and it will use tools from geometric representation theory to study the interrelation between geometry and mathematical physics. Originating from string theory, the project will not only provide a deeper understanding of the universe, it will also train young mathematicians and other highly qualified individuals with the potential to make a significant impact to technology, security, and the economy though their specialised skills.Read moreRead less
Structure of relations: algebra and applications. Relations and relational structures form the fundamental mathematical essence required for studying computational problems and computational systems. This project will provide new algebraic methods for solving old problems in the theory of relations, informing our understanding of computational complexity and the nature of computing.
Australian Laureate Fellowships - Grant ID: FL200100141
Funder
Australian Research Council
Funding Amount
$3,077,547.00
Summary
Real groups and the Langlands program. This program aims to address deep longstanding questions about real groups, algebraic objects which describe the basic symmetries occurring in nature. The study of these basic symmetries is central in all areas of mathematics and they come up in many applications. The expected outcomes include solving a central 50 year old problem of unitarity as well as making major progress in the Langlands program, a grand unification scheme of mathematics. The benefits ....Real groups and the Langlands program. This program aims to address deep longstanding questions about real groups, algebraic objects which describe the basic symmetries occurring in nature. The study of these basic symmetries is central in all areas of mathematics and they come up in many applications. The expected outcomes include solving a central 50 year old problem of unitarity as well as making major progress in the Langlands program, a grand unification scheme of mathematics. The benefits include raising Australia's international research profile, building a large network of international collaboration with top institutions in the world, and increasing capacity in number theory and algebraic geometry, which are playing an ever more important role in technology. Read moreRead less
Coset spaces and Hecke algebra actions. This project will develop fundamental models and their mechanics as tools for studying subtle geometry and hidden symmetry in number systems and systems of equations. These powerful new models will provide an elementary and tractable approach for exploiting patterns that are naturally embedded in complex systems.