Algorithmic and computational advances in geometric group theory. This project aims to combine new algorithmic ideas, high performance computing and experimental mathematics to answer many outstanding questions in the field of geometric group theory. This project will put Australia at the forefront of new computer-assisted research, and give new insights into complex mathematical problems.
Trisections, triangulations and the complexity of manifolds. This project aims at practical representations of 3-dimensional and 4-dimensional spaces as needed in applications. Topology is the mathematical study of the shapes of spaces. Geometry endows spaces with additional structure such as distance, angle and curvature. Special combinatorial structures, such as minimal triangulations, are often closely connected to geometric structures or topological properties. This project aims to construct ....Trisections, triangulations and the complexity of manifolds. This project aims at practical representations of 3-dimensional and 4-dimensional spaces as needed in applications. Topology is the mathematical study of the shapes of spaces. Geometry endows spaces with additional structure such as distance, angle and curvature. Special combinatorial structures, such as minimal triangulations, are often closely connected to geometric structures or topological properties. This project aims to construct computable invariants, connectivity results for triangulations, and algorithms to recognise fundamental topological properties and structures such as trisections and bundles.Read moreRead less
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
Mathematics of Cryptography. The Australian economy and society requires fast, reliable, and secure communication. Current 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 an important and urgent tasks. In addition, t ....Mathematics of Cryptography. The Australian economy and society requires fast, reliable, and secure communication. Current 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 an important and urgent tasks. In addition, 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
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
Inverse problems with partial data. This project aims to use mathematics, in particular the theory of micro-local analysis, to determine the amount of measurements one needs in order to reconstruct an image by some of the tomography methods commonly used in medical imaging. Expected outcomes of this project include showing that an arbitrarily small set of boundary measurements is sufficient to reconstruct the coefficients of various important partial differential equations such as Schrodinger eq ....Inverse problems with partial data. This project aims to use mathematics, in particular the theory of micro-local analysis, to determine the amount of measurements one needs in order to reconstruct an image by some of the tomography methods commonly used in medical imaging. Expected outcomes of this project include showing that an arbitrarily small set of boundary measurements is sufficient to reconstruct the coefficients of various important partial differential equations such as Schrodinger equation, Dirac operators, and Maxwell equations. In addition to providing a theoretical foundation upon which one can build numerical algorithms, this project will also provide the missing link between inverse problems and unique continuation theory. The downstream impact of this research will lead to more efficient and accurate tomography methods which can be implemented in a range of imaging applications.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
Tractable topological computing: Escaping the hardness trap. Computational topology is a young and energetic field that uses computers to solve complex geometric problems driven by pure mathematics, and with diverse applications in biology, signal processing and data mining. A major barrier is that many of these problems are thought to be fundamentally and intractably hard. This project aims to defy such barriers for typical real-world inputs by fusing geometric techniques with technologies from ....Tractable topological computing: Escaping the hardness trap. Computational topology is a young and energetic field that uses computers to solve complex geometric problems driven by pure mathematics, and with diverse applications in biology, signal processing and data mining. A major barrier is that many of these problems are thought to be fundamentally and intractably hard. This project aims to defy such barriers for typical real-world inputs by fusing geometric techniques with technologies from the field of parameterised complexity, creating powerful, practical solutions for these problems. It is expected to shed much-needed light on the vast and puzzling gap between theory and practice, and give researchers fast new software tools for large-scale experimentation and cutting-edge computer proofs.Read moreRead less
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