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
Explicit Construction of Global Function Fields with Many Rational Places. The use of error-correcting codes and cryptosystems is fundamental to the secure and reliable operation of many technological devices that we depend upon in our everyday lives. Essentially invisible, both coding theory and cryptography are essential for banking (ATM machines, e-banking), commerce (e-commerce), defense (cryptography) and entertainment (digital TV and radio, music CDs, DVDs). While certain families of "goo ....Explicit Construction of Global Function Fields with Many Rational Places. The use of error-correcting codes and cryptosystems is fundamental to the secure and reliable operation of many technological devices that we depend upon in our everyday lives. Essentially invisible, both coding theory and cryptography are essential for banking (ATM machines, e-banking), commerce (e-commerce), defense (cryptography) and entertainment (digital TV and radio, music CDs, DVDs). While certain families of "good" codes and cryptosystems can be constructed from specific function fields whose existence is guaranteed by abstract theory, often no actual construction for the function field is currently known. We aim to close this gap, making a greater range of "good" codes and cryptosystems available for practical applications.
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Constructive Module Theory for Algebras. Cohomology is a highly abstract but very powerful tool. To apply it to particular problems, intricate calculations with abstruse objects are required. The project will represent an important step in a long-term research program being undertaken by Cannon and others to develop computational methods that exploit powerful but deeply abstract ideas in algebra and geometry. Solution of the problems will create considerable interest and find application in seve ....Constructive Module Theory for Algebras. Cohomology is a highly abstract but very powerful tool. To apply it to particular problems, intricate calculations with abstruse objects are required. The project will represent an important step in a long-term research program being undertaken by Cannon and others to develop computational methods that exploit powerful but deeply abstract ideas in algebra and geometry. Solution of the problems will create considerable interest and find application in several branches of mathematics including algebraic geometry and algebraic topology. It will be used both to gain theoretical insight and also to solve concrete problems such as determining whether an equation such as x^3+y^9 = z^2 has a solution in integers.Read moreRead less
Constructive Representation Theory and its Applications. The algorithms developed will make it possible to determine the different ways (representations) in which a group of symmetries may be realised as transformations of some space. Such knowledge is required in many areas including differential equations, digital signal processing, engineering ('strut-and-cable' constructions), the design of telephone networks, crystallography and quantum information processing. The high-performance tools fo ....Constructive Representation Theory and its Applications. The algorithms developed will make it possible to determine the different ways (representations) in which a group of symmetries may be realised as transformations of some space. Such knowledge is required in many areas including differential equations, digital signal processing, engineering ('strut-and-cable' constructions), the design of telephone networks, crystallography and quantum information processing. The high-performance tools for linear algebra developed will also find application in cryptography and coding theory. This work represents the latest stage in a long-term project to discover practical algorithms for elucidating the properties of complex algebraic structures - an area where Australia is a world-leader.Read moreRead less
Composition tree algorithms for large matrix groups. This project aims to develop new algorithms for analysing groups. A group is a rather simple mathematical structure – an example is the set of all integers considering only the operations of addition and subtraction. Since the symmetries of an object form a group, groups are ubiquitous throughout mathematics and elsewhere in science. Because it is frequently necessary to determine a group's properties, there is great interest in finding effici ....Composition tree algorithms for large matrix groups. This project aims to develop new algorithms for analysing groups. A group is a rather simple mathematical structure – an example is the set of all integers considering only the operations of addition and subtraction. Since the symmetries of an object form a group, groups are ubiquitous throughout mathematics and elsewhere in science. Because it is frequently necessary to determine a group's properties, there is great interest in finding efficient algorithms for analysing groups. A matrix group is a common type of group whose elements are square matrices. This project plans to employ a novel approach to designing algorithms for analysing large matrix groups, a task which is currently impossible using existing algorithms.Read moreRead less
Computational Methods for Matrix Groups and Group Representations. The symmetry of a system is captured mathematically by the notion of a group. A set of matrices closed under multiplication and the taking of inverses is an important example of a group. For instance, the symmetries of many physical systems and other objects are captured by a group of matrices over the complex numbers. This project will develop the computational tools necessary for constructing and analyzing finite matrix groups ....Computational Methods for Matrix Groups and Group Representations. The symmetry of a system is captured mathematically by the notion of a group. A set of matrices closed under multiplication and the taking of inverses is an important example of a group. For instance, the symmetries of many physical systems and other objects are captured by a group of matrices over the complex numbers. This project will develop the computational tools necessary for constructing and analyzing finite matrix groups over infinite fields such as the complex numbers. These methods will find immediate application to many areas of science and engineering and, in particular, to the theory of quantum computation.
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