Discovery Early Career Researcher Award - Grant ID: DE140100708
Funder
Australian Research Council
Funding Amount
$297,003.00
Summary
Morphing graph drawings. A morphing is a continuous transformation between two drawings of the same topological graph such that at every time instant the drawing has the same topology. Morphings of graph drawings find applications in several areas of computer science, including computer graphics, animation, and modelling. This project will design algorithms for constructing morphings between graph drawings. Unlike any existing method to morph graph drawings, the algorithms designed for this proj ....Morphing graph drawings. A morphing is a continuous transformation between two drawings of the same topological graph such that at every time instant the drawing has the same topology. Morphings of graph drawings find applications in several areas of computer science, including computer graphics, animation, and modelling. This project will design algorithms for constructing morphings between graph drawings. Unlike any existing method to morph graph drawings, the algorithms designed for this project will guarantee bounds on the complexity of the vertex trajectories, guarantee bounds on the resolution of the drawing at every time instant, and deal with topological graphs that are not necessarily planar.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE130100762
Funder
Australian Research Council
Funding Amount
$309,609.00
Summary
The interplay between structures and algorithms in combinatorial optimisation. Networks are ubiquitous in science, technology, and virtually all aspects of life. The project aims to make progress on central questions in the mathematical theory of networks. These include designing efficient algorithms for approximating the Hadwiger number, which is a key measure of the complexity of a network.
Discovery Early Career Researcher Award - Grant ID: DE150100720
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Testing Isomorphism of Algebraic Structures. The algorithmic problem of isomorphism testing seeks to decide whether two objects from a mathematical category are essentially the same. This project focuses on the setting when the categories are from algebra, including but not limited to, groups and polynomials. It is a family of fundamental problems in complexity theory, with important applications in cryptography. The project aims to develop efficient algorithms with provable guarantee, or formal ....Testing Isomorphism of Algebraic Structures. The algorithmic problem of isomorphism testing seeks to decide whether two objects from a mathematical category are essentially the same. This project focuses on the setting when the categories are from algebra, including but not limited to, groups and polynomials. It is a family of fundamental problems in complexity theory, with important applications in cryptography. The project aims to develop efficient algorithms with provable guarantee, or formal hardness proofs, for these problems. Algorithms will be implemented to examine the impacts on certain cryptography schemes. The successful completion of this project will enhance the understanding of computational complexities of these problems, and identify the security of certain cryptography schemes.Read moreRead less
Scalable biocomputing on networks: design and mathematical foundations. This project aims to develop technology with the potential to disrupt computation by providing a way to solve combinatorial mathematical problems in an efficient manner. Electronic computers have revolutionised our lives over the last half-century, but there are tasks they can not do, usually those requiring multi-tasking, much as our brains do. This project aims to overcome some of these problems by physically using molecul ....Scalable biocomputing on networks: design and mathematical foundations. This project aims to develop technology with the potential to disrupt computation by providing a way to solve combinatorial mathematical problems in an efficient manner. Electronic computers have revolutionised our lives over the last half-century, but there are tasks they can not do, usually those requiring multi-tasking, much as our brains do. This project aims to overcome some of these problems by physically using molecular parts of living things moving within specially mathematically designed networks to solve, in parallel, "combinatorial" mathematical problems that vex traditional computers, while using far less energy than electronic devices. This project expects to develop this nascent field into a practically useful, disruptive technology based in Australia.Read moreRead less
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
Topological containment and the Hajós Conjecture: new structure theorems from computer search. This projects aims to characterise when a network contains within it the topology, or shape, of a specific smaller network. It will develop new tools that use computer search to find such characterisations. The outcomes of this project will be used to attack one of the remaining unsolved cases of a famous conjecture dating back over sixty years.
Elliptic curves: number theoretic and cryptographic aspects. Smart information use is of fundamental nature and has a great number of applications. First-generation security solutions are unable to support the modern requirements and new security infrastructures are emerging that must be carefully, but rapidly, defined. This urgently needs new mathematical tools, which is the main goal of this project.
Improved algorithms via random sampling. Randomized methods have recently come into the spotlight when it comes to solving computationally "intractable" subset problems. The running time of a range of algorithms has been improved by replacing their first steps by a simple method of adding to the solution a small subset uniformly at random, and repeating the process many times. This project will explore various other ways how (not necessarily uniform) random sampling can improve the running time ....Improved algorithms via random sampling. Randomized methods have recently come into the spotlight when it comes to solving computationally "intractable" subset problems. The running time of a range of algorithms has been improved by replacing their first steps by a simple method of adding to the solution a small subset uniformly at random, and repeating the process many times. This project will explore various other ways how (not necessarily uniform) random sampling can improve the running time of algorithms; and explore the analysis of polynomial-time randomized algorithms. The project will focus on cycle cutsets and domination problems that have applications in operating systems, chip design and verification, facility location, and surveillance and monitoring.
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New Applications of Additive Combinatorics in Number Theory and Graph Theory. The project aims to advance significantly the interplay between additive combinatorics, number theory and graph theory. The project will use and advance methods and results of additive combinatorics and give new applications to such fundamental problems on Cayley graphs as connectivity, random walks, colouring and dominating sets. The significance of the project is ensured by its goal of advancing existing results and ....New Applications of Additive Combinatorics in Number Theory and Graph Theory. The project aims to advance significantly the interplay between additive combinatorics, number theory and graph theory. The project will use and advance methods and results of additive combinatorics and give new applications to such fundamental problems on Cayley graphs as connectivity, random walks, colouring and dominating sets. The significance of the project is ensured by its goal of advancing existing results and methods of additive combinatorics and also in finding their new applications that have long-lasting impact on paramount problems for Cayley graphs that underlie the architecture of crucial communication networks. Achieving progress on these problems and developing relevant methods of additive combinatorics will be the main outcomes. Read moreRead less
Additive combinatorics, arithmetic algebraic geometry and finite fields. This project aims to combine additive combinatorics and algebraic geometry and apply them to the theory of finite fields. Additive combinatorics and algebraic geometry are mostly developed over the complex numbers and other fields of characteristic zero. This project will bring the power of these different, discrete and continuous areas to finite fields, opening new perspectives for progress on several major problems, inacc ....Additive combinatorics, arithmetic algebraic geometry and finite fields. This project aims to combine additive combinatorics and algebraic geometry and apply them to the theory of finite fields. Additive combinatorics and algebraic geometry are mostly developed over the complex numbers and other fields of characteristic zero. This project will bring the power of these different, discrete and continuous areas to finite fields, opening new perspectives for progress on several major problems, inaccessible by other methods. The project will advance and affect the development of number theory research in Australia and methodologies useful in mathematics and computer science, including cryptography.Read moreRead less