Deep Learning for Graph Isomorphism: Theories and Applications. This project aims to investigate graph isomorphism, a fundamental problem in graph theory, using deep learning techniques. Solutions to graph isomorphism are in demand by researchers in many fields of science, such as biology, chemistry, computer science, and quantum computing. The project expects to advance knowledge about graph isomorphism and state-of-the-art methodologies for its applications. The expected outcomes include new t ....Deep Learning for Graph Isomorphism: Theories and Applications. This project aims to investigate graph isomorphism, a fundamental problem in graph theory, using deep learning techniques. Solutions to graph isomorphism are in demand by researchers in many fields of science, such as biology, chemistry, computer science, and quantum computing. The project expects to advance knowledge about graph isomorphism and state-of-the-art methodologies for its applications. The expected outcomes include new theoretical insights on combinatorial structures of graphs, efficient heuristic techniques for (maximum) subgraph isomorphism, and structured representation learning. The project should provide significant benefits to research in a wide range of science fields, as well as many real-world applications.Read moreRead less
Mathematics of Elliptic Curve Cryptography. The Australian society and economy requires fast, reliable, and secure digital infrastructure. First-generation security solutions cannot support the efficiency and scalability requirements of wireless and embedded consumer applications. New security infrastructures are emerging and must be carefully, but rapidly, defined and analysed. Thus developing a new framework in this area is one of the most important and urgent tasks. Besides, the intended wor ....Mathematics of Elliptic Curve Cryptography. The Australian society and economy requires fast, reliable, and secure digital infrastructure. First-generation security solutions cannot support the efficiency and scalability requirements of wireless and embedded consumer applications. New security infrastructures are emerging and must be carefully, but rapidly, defined and analysed. Thus developing a new framework in this area is one of the most important and urgent tasks. Besides, the intended work advances our knowledge of the theory and the quality of our culture. As such, it will promote the Australian science and will also have many practical applications in Computer Security and E-Commerce.Read moreRead less
Continued Fractions and Torsion on Hyperelliptic Curves. Scientific advance should not blindly add to our knowledge; a true advance brings insights that collapse different issues into one. Understanding more is to need to remember less. For an important class of examples, this project identifies the study of a fundamental invariant of a quadratic number field, its regulator and hence its class number, with maximum torsion on the Jacobian variety of an hyperelliptic curve. The investigator's meth ....Continued Fractions and Torsion on Hyperelliptic Curves. Scientific advance should not blindly add to our knowledge; a true advance brings insights that collapse different issues into one. Understanding more is to need to remember less. For an important class of examples, this project identifies the study of a fundamental invariant of a quadratic number field, its regulator and hence its class number, with maximum torsion on the Jacobian variety of an hyperelliptic curve. The investigator's methods will surprise some longstanding problems into submission and in particular will lead them to reveal full data on torsion on hyperelliptic curves of low genus.
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Number Theoretic Methods in Cryptography. It is well known that Number Theory, besides its intrinsic beauty, provides many powerful tools for modern Cryptography. The aim of the project is to formulate and solve new and important mathematical problems, which lie in the background of modern cryptography. They are also of independent value for pure mathematics because they very often stimulate new approaches to and new surprising points of view on classical results and methods. The main outcome w ....Number Theoretic Methods in Cryptography. It is well known that Number Theory, besides its intrinsic beauty, provides many powerful tools for modern Cryptography. The aim of the project is to formulate and solve new and important mathematical problems, which lie in the background of modern cryptography. They are also of independent value for pure mathematics because they very often stimulate new approaches to and new surprising points of view on classical results and methods. The main outcome will be advancing our theoretical knowledge about several major cryptosystems. The project will extend and enrich the area of applications of mathematics to cryptography and related areas.Read moreRead less
A new generation of fractals: theory, computation, and applications particularly to digital imaging. The project develops the mathematical and algorithmic foundations of superfractals and applies these results to a number of different areas, including in particular, digital imaging. For example, the ``third generation'' of mobile communications (3G), combines wireless mobile technology with high data transmission capacities. Currently the requirement for extensive bandwidth is a problem for e ....A new generation of fractals: theory, computation, and applications particularly to digital imaging. The project develops the mathematical and algorithmic foundations of superfractals and applies these results to a number of different areas, including in particular, digital imaging. For example, the ``third generation'' of mobile communications (3G), combines wireless mobile technology with high data transmission capacities. Currently the requirement for extensive bandwidth is a problem for efficient use. Superfractals and the associated colouring algorithm could be used to develop a new system to produce synthetic content for wireless devices that would require only low bandwidth.Read moreRead less
Mathematics of Cryptography. The Australian society and economy requires fast, reliable, and secure communication. First-generation security solutions are not capable of supporting the efficiency and scalability requirements of mass-market adoption of wireless and embedded consumer applications. New security infrastructures are emerging and must be carefully, but rapidly, defined. Thus developing new mathematically solid tools in this area is one of the most important and urgent tasks. Besides, ....Mathematics of Cryptography. The Australian society and economy requires fast, reliable, and secure communication. First-generation security solutions are not capable of supporting the efficiency and scalability requirements of mass-market adoption of wireless and embedded consumer applications. New security infrastructures are emerging and must be carefully, but rapidly, defined. Thus developing new mathematically solid tools in this area is one of the most important and urgent tasks. Besides, the intended work advances our knowledge of the theory and the quality of our culture. As such, it will promote the Australian science and will also have many practical applications in Cryptography, Computer Security and E-Commerce.Read moreRead less
Ring constructions and algorithms for enhancing performance of BCH codes. BCH codes form a major class of codes used in modern communication systems. The aim of this project is to enhance the efficiency of this class of codes by combining them in constructions enabling correction of deletion and insertion errors, and develop efficient implementations of encoding and decoding algorithms incorporating soft decision methods for enhanced error correction. Significance of the project is explained by ....Ring constructions and algorithms for enhancing performance of BCH codes. BCH codes form a major class of codes used in modern communication systems. The aim of this project is to enhance the efficiency of this class of codes by combining them in constructions enabling correction of deletion and insertion errors, and develop efficient implementations of encoding and decoding algorithms incorporating soft decision methods for enhanced error correction. Significance of the project is explained by the role of fast, secure and reliable communications in modern information and communication technology. Expected outcomes include new efficient algorithms and commercial modules available for symbolic computation systems with applications in telecommunications industry.
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Multiplicative structure of rational functions. This project aims to develop new methods of investigating fundamental number theoretic notions of torsion and multiplicative dependence between objects of great interest such as rational functions and their values. This includes investigating such celebrated objects as torsion points on elliptic curves and torsion subgroups on algebraic varieties. The goal is to develop new methods and make pivotal advances towards solving several fundamental probl ....Multiplicative structure of rational functions. This project aims to develop new methods of investigating fundamental number theoretic notions of torsion and multiplicative dependence between objects of great interest such as rational functions and their values. This includes investigating such celebrated objects as torsion points on elliptic curves and torsion subgroups on algebraic varieties. The goal is to develop new methods and make pivotal advances towards solving several fundamental problems where multiplicative dependence plays a crucial role. The expected outcome is to provide deeper understanding of the intriguing nature of torsion and multiplicative dependence and thus open new perspectives for their applications in number theory and beyond.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE120100040
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Partitioning and ordering Steiner triple systems. Steiner triple systems are fundamental mathematical objects with many real-world applications. This project will develop deep new insights into these objects, resulting in systems allowing many users to simultaneously use a communication channel, and in schemes for preventing the loss of computer data due to hard disk failures.
From quantum integrable systems to algebraic geometry and combinatorics. The purpose of this project is to investigate the deep connections that have recently emerged between the study of an area of mathematical physics (quantum integrable systems) and subjects of pure mathematics (enumerative and algebraic combinatorics, and algebraic geometry). These connections have a common root, which this project plans to reveal using novel methods coming from quantum integrability. This approach is expect ....From quantum integrable systems to algebraic geometry and combinatorics. The purpose of this project is to investigate the deep connections that have recently emerged between the study of an area of mathematical physics (quantum integrable systems) and subjects of pure mathematics (enumerative and algebraic combinatorics, and algebraic geometry). These connections have a common root, which this project plans to reveal using novel methods coming from quantum integrability. This approach is expected to illuminate these subjects leading to a new unified and interdisciplinary picture, and to resolve important open problems in the study of certain algebraic varieties and of their cohomology in the theory of symmetric functions, and related counting problems.Read moreRead less