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Algebras with Frobenius morphisms and quantum groups. In this digitalized world, our life relies on mathematics more than ever. Counting and numbers are just one example of this. Another is the public key codes for online payments and transactions. Mathematics is of enormous importance in this technology dominated age. This proposal is to carry out high level mathematical research in Australia. Basic research on quantum groups underpins applied research and certain areas such as quantum mechanic ....Algebras with Frobenius morphisms and quantum groups. In this digitalized world, our life relies on mathematics more than ever. Counting and numbers are just one example of this. Another is the public key codes for online payments and transactions. Mathematics is of enormous importance in this technology dominated age. This proposal is to carry out high level mathematical research in Australia. Basic research on quantum groups underpins applied research and certain areas such as quantum mechanics and string theory. Some structure of quantum groups is too complicated to be seen by even a professional mathematician. A possible interpretation by using representations over a finite field would make it more usable and accessible by computer.Read moreRead less
Modular representations of cyclotomic algebras. This project addresses cutting edge questions in the representation theory of cyclotomic Hecke algebras. Our main focus will be computing decomposition matrices for these algebras. We approach this question from several different directions, each of which will give new insights and lead to significant advances in the theory. The decomposition number problem is important because its' solution gives deep structural information about these algebras wh ....Modular representations of cyclotomic algebras. This project addresses cutting edge questions in the representation theory of cyclotomic Hecke algebras. Our main focus will be computing decomposition matrices for these algebras. We approach this question from several different directions, each of which will give new insights and lead to significant advances in the theory. The decomposition number problem is important because its' solution gives deep structural information about these algebras which can then be applied in other areas. This project will have high impact because cyclotomic Hecke algebras have applications in many different areas and they are currently a hot topic of research in mathematics.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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Invariant theory, cellularity and geometry. Mathematics underpins every aspect of people's interactions with nature (e.g. physics) and with each other (e.g. finance). Its uses range from formulating physical laws in order to understand and predict nature, to analysis of financial concepts and transactions. This project will make fundamental contributions to the mathematics of symmetry. Benefits include enhancement of Australia's position at the very frontier of world class mathematical research, ....Invariant theory, cellularity and geometry. Mathematics underpins every aspect of people's interactions with nature (e.g. physics) and with each other (e.g. finance). Its uses range from formulating physical laws in order to understand and predict nature, to analysis of financial concepts and transactions. This project will make fundamental contributions to the mathematics of symmetry. Benefits include enhancement of Australia's position at the very frontier of world class mathematical research, and a myriad of potential applications to physics, coding theory, information technology, electronic security and experimental design.Read moreRead less
Braid monoids, presentations and normal forms. Braid groups arise naturally in various areas of mathematics, physics and computer science including knot theory, Lie theory, quantum groups and cryptography. There is a uniform geometric description of braid groups; however this is not the case algebraically. This project aims to find the connections between the algebra, combinatorics and geometry of braid groups in order to obtain a uniform algebraic description. This generalisation will allow adv ....Braid monoids, presentations and normal forms. Braid groups arise naturally in various areas of mathematics, physics and computer science including knot theory, Lie theory, quantum groups and cryptography. There is a uniform geometric description of braid groups; however this is not the case algebraically. This project aims to find the connections between the algebra, combinatorics and geometry of braid groups in order to obtain a uniform algebraic description. This generalisation will allow advances in the related areas of mathematics and physics. In addition to theoretical results, new algorithms for calculating in braid groups will be given, which can then be implemented computationally.
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Affine buildings and Hecke algebras. This project is breakthrough science. Affine buildings, Ramanujan complexes and the representation theory of affine Hecke algebras are on the cutting edge of research in mathematics, as evidenced by recent special programs at Cambridge and Princeton. The outcomes from this project will be published in first class journals and they will be implemented in computer algebra systems for world wide application. The project is likely to have flow-on effects in othe ....Affine buildings and Hecke algebras. This project is breakthrough science. Affine buildings, Ramanujan complexes and the representation theory of affine Hecke algebras are on the cutting edge of research in mathematics, as evidenced by recent special programs at Cambridge and Princeton. The outcomes from this project will be published in first class journals and they will be implemented in computer algebra systems for world wide application. The project is likely to have flow-on effects in other disciplines, notably communication networks, mathematical physics and computer science.
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Hecke Algebras in Algebra and Analysis. The aim of this program is to adapt techniques from harmonic analysis and operator-algebraic representation theory to study Hecke algebras arising in algebraic and geometric settings. The relevant analytic structures are C*-algebras and the fundamental question is then "Which Hecke algebras have a faithful enveloping C*-algebra?" We investigate this question, first by developing an appropriate theory of crossed products by semigroups and, second, by using ....Hecke Algebras in Algebra and Analysis. The aim of this program is to adapt techniques from harmonic analysis and operator-algebraic representation theory to study Hecke algebras arising in algebraic and geometric settings. The relevant analytic structures are C*-algebras and the fundamental question is then "Which Hecke algebras have a faithful enveloping C*-algebra?" We investigate this question, first by developing an appropriate theory of crossed products by semigroups and, second, by using the notion of topologization which enables the Hecke algebra to be studied in the context of topological groups.Read moreRead less
Constructive Module Theory for Algebras. Cohomology is a highly abstract but very powerful tool. To apply it to particular problems, intricate calculations with abstruse objects are required. The project will represent an important step in a long-term research program being undertaken by Cannon and others to develop computational methods that exploit powerful but deeply abstract ideas in algebra and geometry. Solution of the problems will create considerable interest and find application in seve ....Constructive Module Theory for Algebras. Cohomology is a highly abstract but very powerful tool. To apply it to particular problems, intricate calculations with abstruse objects are required. The project will represent an important step in a long-term research program being undertaken by Cannon and others to develop computational methods that exploit powerful but deeply abstract ideas in algebra and geometry. Solution of the problems will create considerable interest and find application in several branches of mathematics including algebraic geometry and algebraic topology. It will be used both to gain theoretical insight and also to solve concrete problems such as determining whether an equation such as x^3+y^9 = z^2 has a solution in integers.Read moreRead less
Quantized representation theory. The representation theory of quantized algebras, or deformation algebras, is a rapidly expanding and exciting field. It has wide ranging applications from within mathematics, to knot theory and statistical mechanics. This project addresses several important open problems in the area with an emphasis on structural innovations and computing explicit numerical invariants.
Geometric structures in representation theory. Mathematics underpins every aspect of people's interactions with nature (e.g. physics) and with each other (e.g. finance). Its uses range from formulating physical laws in order to understand and predict nature, to analysis of financial concepts and transactions. This project will formulate and develop three new fundamental mathematical concepts: cellular algebras, eigenspace geometries, and diagram algebras. Benefits include enhancement of Australi ....Geometric structures in representation theory. Mathematics underpins every aspect of people's interactions with nature (e.g. physics) and with each other (e.g. finance). Its uses range from formulating physical laws in order to understand and predict nature, to analysis of financial concepts and transactions. This project will formulate and develop three new fundamental mathematical concepts: cellular algebras, eigenspace geometries, and diagram algebras. Benefits include enhancement of Australia's position at the very frontier of world class mathematical research, and a myriad of potential applications to physics, coding theory, information technology, electronic security and experimental design.Read moreRead less