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
Combinatorial structures for computer security and communication. Hadamard matrices in their various guises arise many times in the study of reliable communications and secure communications. The aim of this research project is to use the theory of cyclotomy in both fields and rings to find new number theoretic results which will then be used to obtain new with zero or small autocorrelation functions.
The significance of this research is to propose new construction of Hadamard matrices and bloc ....Combinatorial structures for computer security and communication. Hadamard matrices in their various guises arise many times in the study of reliable communications and secure communications. The aim of this research project is to use the theory of cyclotomy in both fields and rings to find new number theoretic results which will then be used to obtain new with zero or small autocorrelation functions.
The significance of this research is to propose new construction of Hadamard matrices and block designs for computer security and wireless network communication.
We expect some new classes of Hadamard matrices and block designs can be constructed for security and communication applications and several papers would be submitted or published.
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Representations of dynamical systems, amenability, and proper actions. Mathematicians study abstract objects by representing them in terms of well-understood concrete models, and need to know when a representation is faithful, in the sense that the model contains complete information. Dynamical systems are an abstraction of physical systems suitable for studying time evolution and symmetries. The project aims to determine when important representations of dynamical systems are faithful, or, in ....Representations of dynamical systems, amenability, and proper actions. Mathematicians study abstract objects by representing them in terms of well-understood concrete models, and need to know when a representation is faithful, in the sense that the model contains complete information. Dynamical systems are an abstraction of physical systems suitable for studying time evolution and symmetries. The project aims to determine when important representations of dynamical systems are faithful, or, in mathematical language, when the dynamical system is amenable. The proposed strategy involves extending Rieffel's notion of proper actions; the construction should be of wide applicability apart from the intended applications to amenability.Read moreRead less
Totally disconnected groups, representations and discrete mathematics. This project involves participation in programs at the Institute of Advanced Studies in Princeton and the nearby Center for Discrete Mathematics and Theoretical Computer Science that are designed to initiate collaborations across distinct mathematical research areas. These programs will set future research directions and could lead to innovations in computer science. Discoveries I have made in one of the research areas mean ....Totally disconnected groups, representations and discrete mathematics. This project involves participation in programs at the Institute of Advanced Studies in Princeton and the nearby Center for Discrete Mathematics and Theoretical Computer Science that are designed to initiate collaborations across distinct mathematical research areas. These programs will set future research directions and could lead to innovations in computer science. Discoveries I have made in one of the research areas mean that I may be able to make substantial contributions to these programs. Early involvement in influential programs such as these means that Australia is well placed to take advantage of developments that result and also enhances the reputation of Australian mathematics.Read moreRead less
The structure of quantum groups. We propose to study the structure of mathematical objects used in describing symmetries of micro-scale phenomena. The project will significantly develop already well established Australian-Korean cooperation in this exciting and rapidly growing area of research. The results will be immediately applicable to related fields of mathematics, most notably to noncommutative geometry. In the long run, the outcomes will help in better understanding of fundamental problem ....The structure of quantum groups. We propose to study the structure of mathematical objects used in describing symmetries of micro-scale phenomena. The project will significantly develop already well established Australian-Korean cooperation in this exciting and rapidly growing area of research. The results will be immediately applicable to related fields of mathematics, most notably to noncommutative geometry. In the long run, the outcomes will help in better understanding of fundamental problems of modern quantum physics.Read moreRead less
Ergodic theory and number theory. Recent advances in the theory of measured dynamical systems investigated by the proponents include new versions of entropy, and the study of spectral theory for non-singular systems. These will be further developed in this joint project with the French CNRS. The results are expected to have interesting applications in physics and number theory.
Boundary Value Problems for Differential Inclusions. Boundary value problems (BVPs) for differential inclusions are mathematical equations that accurately describe the complex world around us. This project aims to answer important mathematical questions such as:
(i) Under what conditions do BVPs for differential inclusions actually have solutions?
(ii) If solutions do exist, what are their properties?
(iii) If solutions are too complicated to be found explicitly, then how can they be approxim ....Boundary Value Problems for Differential Inclusions. Boundary value problems (BVPs) for differential inclusions are mathematical equations that accurately describe the complex world around us. This project aims to answer important mathematical questions such as:
(i) Under what conditions do BVPs for differential inclusions actually have solutions?
(ii) If solutions do exist, what are their properties?
(iii) If solutions are too complicated to be found explicitly, then how can they be approximated?
The approach is to use modern tools from mathematical analysis, including new differential inequalities.
The project outcomes will:
(a) Significantly advance mathematical knowledge for differential inclusions
(b) Have many applications to areas of science, engineering and technology.Read moreRead less
Entropy and maximal entropy in Markov systems. Entropy is a measure of how well-ordered a system is: chaotic systems have high entropy. Two approaches to entropy are available, via the limiting behaviour of the orbits of points, which yields topological entropy, and via the behaviour of the distributions of measures of partitions, yielding measure-theoretic entropy. The topological entropy is the least upper bound of entropies of all possible measures. We study when there is a measure which real ....Entropy and maximal entropy in Markov systems. Entropy is a measure of how well-ordered a system is: chaotic systems have high entropy. Two approaches to entropy are available, via the limiting behaviour of the orbits of points, which yields topological entropy, and via the behaviour of the distributions of measures of partitions, yielding measure-theoretic entropy. The topological entropy is the least upper bound of entropies of all possible measures. We study when there is a measure which realises this bound, describing the structure of such systems via Markov and Bratteli diagrams. Our methods will be applied to new versions of entropy for non-singular systems. This will assist in the description of chaotic behaviour.Read moreRead less
Lie-type methods for totally disconnected groups. Groups are algebraic objects which convey symmetry, much as numbers convey size. For example, the rotations of a sphere form a group. This rotation group is one of a class known as the Lie groups that is well understood and has important applications. Totally disconnected groups arise as symmetries of network structures having nodes and a `neighbour' relation between nodes. The Australian investigator has discovered powerful methods for analysing ....Lie-type methods for totally disconnected groups. Groups are algebraic objects which convey symmetry, much as numbers convey size. For example, the rotations of a sphere form a group. This rotation group is one of a class known as the Lie groups that is well understood and has important applications. Totally disconnected groups arise as symmetries of network structures having nodes and a `neighbour' relation between nodes. The Australian investigator has discovered powerful methods for analysing totally disconnected groups which have parallels with Lie group techniques. This project will develop these parallels and establish links with international researchers on Lie groups.Read moreRead less