A new model for random discrete structures: distributions, counting and sampling. Random discrete structures are used in countless applications across science for modelling complex systems. This project will study a new, very general model of random discrete structures which encapsulates both random networks and random matrices. This project will develop general tools for working with this model, thereby unlocking the model for use by practitioners in areas such as physics, biology, statistics a ....A new model for random discrete structures: distributions, counting and sampling. Random discrete structures are used in countless applications across science for modelling complex systems. This project will study a new, very general model of random discrete structures which encapsulates both random networks and random matrices. This project will develop general tools for working with this model, thereby unlocking the model for use by practitioners in areas such as physics, biology, statistics and cryptography. The questions that will be tackled are fundamental problems in probability, and include as special cases the analysis of subgraph distribution in models of random networks, and the joint distribution of entries of contingency tables, which are important in statistics.Read moreRead less
Towards the prime power conjecture. This project attacks a famous and long standing conjecture in pure mathematics that has important ramifications in many applied areas. The project aims to determine when it is possible to produce more efficient codes for electronic communication and statistically balanced designs for experiments in areas as diverse as agriculture and psychology.
Australian Laureate Fellowships - Grant ID: FL120100125
Funder
Australian Research Council
Funding Amount
$1,796,966.00
Summary
Advances in the analysis of random structures and their applications. This project will provide new approaches, insights and results for probabilistic combinatorics. This area has contributed in exciting ways elsewhere in mathematics and provides versatile tools of widespread use in algorithmic computer science, with other applications in physics, coding theory for communications, and genetics.
A new approach to compressed sensing. Compressed sensing is an exciting new paradigm promising vastly improved signal sampling and reconstruction in a wide variety of applications including digital cameras, mobile phones and MRI machines. This project will explore a newly discovered approach to compressed sensing which uses mathematical arrays known as hash families.
Expander graphs, isoperimetric numbers, and forwarding indices. Expanders are sparse but well connected networks. With numerous applications to modern technology, they have attracted many world leaders in mathematics and computer science. This project aims at substantial advancement on some important problems on expanders and related areas. It will put Australia at the forefront of this topical field.
Discovery Early Career Researcher Award - Grant ID: DE190100888
Funder
Australian Research Council
Funding Amount
$333,924.00
Summary
Linear recurrence sequences over function fields and their applications. This project aims to deeply and systematically develop the theory of linear recurrence sequences (LRS) defined over function fields. Linear recurrence sequences (LRS) appear almost everywhere in mathematics and computer science. The project is expected to expand our knowledge on LRS and will span a wide range of new research directions. Through investigating and revealing the theoretical and practical aspects of LRS over fu ....Linear recurrence sequences over function fields and their applications. This project aims to deeply and systematically develop the theory of linear recurrence sequences (LRS) defined over function fields. Linear recurrence sequences (LRS) appear almost everywhere in mathematics and computer science. The project is expected to expand our knowledge on LRS and will span a wide range of new research directions. Through investigating and revealing the theoretical and practical aspects of LRS over function fields, the project will enrich the toolkits for cybersecurity by providing new approaches to cryptography. The outcomes of the project will help position Australia as a leader in this field.Read moreRead less
Fundamental mathematical structures in statistical and quantum systems. Mathematics is playing a key role in modern science and technology. This project will bring together world leading experts from Australia and the USA to unravel the most fundamental mathematical structures in of statistical and quantum systems arising in settings ranging from physics of tiny quantum dots to string theory in high energy physics. This research will ensure Australia's involvement in cutting-edge international d ....Fundamental mathematical structures in statistical and quantum systems. Mathematics is playing a key role in modern science and technology. This project will bring together world leading experts from Australia and the USA to unravel the most fundamental mathematical structures in of statistical and quantum systems arising in settings ranging from physics of tiny quantum dots to string theory in high energy physics. This research will ensure Australia's involvement in cutting-edge international developments in mathematical sciences poised to deliver new significant results in the fundamental quantum theory of matter. The project will also contribute to training young researchers to maintain Australia's international standing in fundamental science.Read moreRead less
Quantization of polyhedral surfaces. Recent developments in the theory of discrete surfaces have revealed their fascinating links to many other areas of mathematics including integrable systems and quantum geometry. Rapid progress in this field is motivated by applications in pure mathematics, mathematical physics, computer graphics and engineering. Australian researchers are world recognized experts in integrable systems and this project will link them together with German experts in discrete d ....Quantization of polyhedral surfaces. Recent developments in the theory of discrete surfaces have revealed their fascinating links to many other areas of mathematics including integrable systems and quantum geometry. Rapid progress in this field is motivated by applications in pure mathematics, mathematical physics, computer graphics and engineering. Australian researchers are world recognized experts in integrable systems and this project will link them together with German experts in discrete differential geometry. The project will advance our knowledge base in fundamental and applied sciences and offer a unique research training opportunity for students in contemporary areas of pure and applied mathematics.Read moreRead less
Diffusion driven pattern formation and signal propagation in spatially complex excitable media. A basic understanding of the mechanisms for pattern formation, from the spots on leopards to electrical signalling of neurons, has been achieved through reaction-diffusion equations. However to obtain a complete understanding, which is vital for many applications, it is necessary to modify this mathematical model to incorporate spatial complexities in the underlying media. This project will develop ....Diffusion driven pattern formation and signal propagation in spatially complex excitable media. A basic understanding of the mechanisms for pattern formation, from the spots on leopards to electrical signalling of neurons, has been achieved through reaction-diffusion equations. However to obtain a complete understanding, which is vital for many applications, it is necessary to modify this mathematical model to incorporate spatial complexities in the underlying media. This project will develop a fractional calculus framework for pattern formation, including signal propagation, in spatially complex and excitable media. In a particular application we will model the way in which the signalling properties of neurons depend critically on their spatial complexity.Read moreRead less
New mathematics of fractional diffusion for understanding cognitive impairment at the neuronal level. As Australia's population ages, cognitive impairment due to cortical ageing and neurodegeneration is looming as the nation's greatest health problem. The project will deliver new, more realistic, mathematical models for a mechanistic understanding of cognitive impairment at the neuronal level. This understanding is a vital first step in targeting drugs, e.g., to influence neuronal spine proper ....New mathematics of fractional diffusion for understanding cognitive impairment at the neuronal level. As Australia's population ages, cognitive impairment due to cortical ageing and neurodegeneration is looming as the nation's greatest health problem. The project will deliver new, more realistic, mathematical models for a mechanistic understanding of cognitive impairment at the neuronal level. This understanding is a vital first step in targeting drugs, e.g., to influence neuronal spine properties, for preventative health care. The project will maintain international collaborations, between applied mathematicians at UNSW, Sydney and biomathematicians and neuroscientists at Mount Sinai School of Medicine, New York, providing ongoing training opportunities for Australian scientists in this cutting edge biomathematical research.Read moreRead less