From individuals to mass organisation: aggregation, synchronisation and collective movement in locusts. By combining field biology, robotics and mathematics, this project will determine how animals flock or swarm and, in particular, how locust nymphs control their collective movement over their lifetime. The mathematical models derived during the project will be directly applied to controlling outbreaks of locusts in Australia, South and North Africa.
Algebraic categories and categorical algebra. Algebra is the study of operations, such as addition and multiplication, and the relationships between these operations. This project will study two exciting new branches of algebra, quantum algebra and postmodern algebra, which will lead to important advances in physics, geometry, and computing.
Foundations of quantum cryptography for distribution of secret keys. Quantum cryptographic systems have the advantage of mathematically provable security and privacy, addressing security threats to communications as information and communications technologies proliferate. This project aims to quantify a quantum channel's capability for secure communications. This quantity provides the ultimate limit to benchmark practical quantum key distribution protocols for their performance. This will signif ....Foundations of quantum cryptography for distribution of secret keys. Quantum cryptographic systems have the advantage of mathematically provable security and privacy, addressing security threats to communications as information and communications technologies proliferate. This project aims to quantify a quantum channel's capability for secure communications. This quantity provides the ultimate limit to benchmark practical quantum key distribution protocols for their performance. This will significantly advance the theory of quantum cryptography and knowledge of the fundamental resource of secret keys. It is expected to have immediate application for the classical security of existing (non-quantum) communication devices, and benefit security, military, government, industry, individuals, and the community.Read moreRead less
A mathematical foundation and novel solutions for highly secure communications. This project will deliver novel solutions to security and privacy in communication networks by exploring the power of quantum information with mathematical tools from operator stuctures. It will significantly advance our knowledge about quantum communication, with expected benefits on Australian social, economic and even military security.
Frontiers in Data Science: Analysing Distributions as Data. This project aims to develop the statistical foundations of a new approach to analysing large and complex data, based on building distributional approximations of the data, which can then be analysed by standard statistical methods. The need to analyse very large and complex datasets has become a vital part of everyday life, particularly in the analysis of national problems in public health, environmental pollution, computer network sec ....Frontiers in Data Science: Analysing Distributions as Data. This project aims to develop the statistical foundations of a new approach to analysing large and complex data, based on building distributional approximations of the data, which can then be analysed by standard statistical methods. The need to analyse very large and complex datasets has become a vital part of everyday life, particularly in the analysis of national problems in public health, environmental pollution, computer network security and climate extremes. The project expects to change our way of thinking in how to be smarter about what data we use (and collect) for analysis, rather than relying on brute force analysis of large datasets. The project is expected to transform the knowledge base of the discipline, and the resulting techniques will enable across-the-board research advances for many industries and disciplines.Read moreRead less
New statistical methods for identifying micro-ribonucleic acid (miRNA) regulatory networks. Understanding gene regulatory networks is critical in the understanding of fundamental biological systems. These networks have important implications for the discovery of fundamental mechanisms relating to the diagnosis and management of many illnesses. This research will provide new statistical methods to identify regulatory micro-ribonucleic acid modules and to understand their relationship in gene regu ....New statistical methods for identifying micro-ribonucleic acid (miRNA) regulatory networks. Understanding gene regulatory networks is critical in the understanding of fundamental biological systems. These networks have important implications for the discovery of fundamental mechanisms relating to the diagnosis and management of many illnesses. This research will provide new statistical methods to identify regulatory micro-ribonucleic acid modules and to understand their relationship in gene regulatory networks through multiple covariance estimation and multivariate classification techniques. My results should enable researchers to better understand the regulation underlying biological systems, leading to improved human health, medical and biological research outcomes.Read moreRead less
Counting points on algebraic surfaces. This project aims to develop algorithms for calculating the number of solutions to polynomial equations and to compute zeta functions of certain types of algebraic varieties. Existing algorithms cannot solve these problems. The new algorithms will enable researchers in number theory to test and refine conjectures on generalisations of many famous problems, such as the Sato-Tate conjecture, the Lang-Trotter conjecture and the Birch-Swinnerton-Dyer conjecture ....Counting points on algebraic surfaces. This project aims to develop algorithms for calculating the number of solutions to polynomial equations and to compute zeta functions of certain types of algebraic varieties. Existing algorithms cannot solve these problems. The new algorithms will enable researchers in number theory to test and refine conjectures on generalisations of many famous problems, such as the Sato-Tate conjecture, the Lang-Trotter conjecture and the Birch-Swinnerton-Dyer conjecture. The project will also have a flow-on effect in other areas of mathematics and computer science where zeta functions play a central role, including cryptography, coding theory and mathematical physics.Read moreRead less
The Spectral Theory and Harmonic Analysis of Geometric Differential Operators. The project will involve mathematical research of the highest international standard in two very active and far-reaching field of mathematics: quantum chaos, and harmonic analysis. Progress in these fields will have implications in areas such as communications technology (e.g. image compression), quantum theory, and mathematical analysis (e.g. partial differential equations).
Interactions between geometric and topological structures. This project aims to develop the necessary tools for a geometric imagination in higher dimensions and to bridge the gap between low and high dimensions. Topology is the mathematical study of the shape of spaces. Geometry endows spaces with additional structure such as distance, angle and curvature. Moduli space encodes the different ways in which a geometry can be assigned to a space. Expected outcomes of the project include construction ....Interactions between geometric and topological structures. This project aims to develop the necessary tools for a geometric imagination in higher dimensions and to bridge the gap between low and high dimensions. Topology is the mathematical study of the shape of spaces. Geometry endows spaces with additional structure such as distance, angle and curvature. Moduli space encodes the different ways in which a geometry can be assigned to a space. Expected outcomes of the project include construction of computable invariants, solution of realisation problems and understanding degeneration of geometries.Read moreRead less
Operator algebras as models for dynamics and geometry. Operator algebra is the mathematical theory which describes quantum physics and predicts how quantum systems will behave. Through this project, the researcher's recent discoveries in operator algebra will give us new insight into the dynamics and geometry - that is, the behaviour and shape - of the quantum world.