Asymptotic Expansions and Large Deviations in Probability and Statistics: Theory and Applications. Statistics is the major enabling science in a number of disciplines. This is fundamental research in probability and statistics but it has wide applications in Biology and Social Sciences which will ultimately be of national benefit. The behaviour of self normalized sums is an exciting new area of fundamental research that has implications for the application of statistics in many areas. U-statist ....Asymptotic Expansions and Large Deviations in Probability and Statistics: Theory and Applications. Statistics is the major enabling science in a number of disciplines. This is fundamental research in probability and statistics but it has wide applications in Biology and Social Sciences which will ultimately be of national benefit. The behaviour of self normalized sums is an exciting new area of fundamental research that has implications for the application of statistics in many areas. U-statistics for dependent situations has direct application to understanding financial time series and the analysis of sample survey data. Saddlepoint methods provide extremely accurate approximations in a number of important applications.
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Empirical saddlepoint approximations and self-normalized limit theorems. Finite population sampling and resampling methods such as the bootstrap and randomization methods are central in a number of areas of application and M-estimates are the major method used to give robust methods under mild conditions; in both these areas statistics are used which are Studentized or self-normalized. We will develop asymptotic approaches for such statistics. Saddlepoint and empirical saddlepoint methods will ....Empirical saddlepoint approximations and self-normalized limit theorems. Finite population sampling and resampling methods such as the bootstrap and randomization methods are central in a number of areas of application and M-estimates are the major method used to give robust methods under mild conditions; in both these areas statistics are used which are Studentized or self-normalized. We will develop asymptotic approaches for such statistics. Saddlepoint and empirical saddlepoint methods will be used to give methods which have second order relative accuracy in large deviation regions and we will obtain limit results and Edgeworth approximations. Emphasis will be on obtaining results under weak conditions necessary for applications.Read moreRead less
Chromatic polynomials, random graphs, and error-correcting codes: a unified approach to graph colouring problems. Through a unified approach involving cutting-edge results on chromatic polynomials, random graphs, matroids, and error-correcting codes, this project will establish the foundations for a rigorous mathematical framework for attempting to provide a short, transparent and illuminating solution to the Four Colour Problem. The project will support developments in computer science and sta ....Chromatic polynomials, random graphs, and error-correcting codes: a unified approach to graph colouring problems. Through a unified approach involving cutting-edge results on chromatic polynomials, random graphs, matroids, and error-correcting codes, this project will establish the foundations for a rigorous mathematical framework for attempting to provide a short, transparent and illuminating solution to the Four Colour Problem. The project will support developments in computer science and statistical mechanics and is likely to have flow-on effects in real-world disciplines such as network communication. This project will also strengthen Australia's international presence in discrete mathematics and will further strengthen ties between Australian and international mathematicians.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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p-Adic Methods in Arithmetic Geometry. This project concerns algorithms for determining the number of
solutions to systems of polynomial equations over finite fields
by p-adic methods. Our goal is to determine a fundamental
invariant, the zeta function, appearing in arithmetic geometry,
whose characterization was the subject of the famous Weil
conjectures.
....p-Adic Methods in Arithmetic Geometry. This project concerns algorithms for determining the number of
solutions to systems of polynomial equations over finite fields
by p-adic methods. Our goal is to determine a fundamental
invariant, the zeta function, appearing in arithmetic geometry,
whose characterization was the subject of the famous Weil
conjectures.
We seek to understand and develop p-adic methods for determining
zeta functions, taking as point of departure the methods of Satoh
and Mestre for elliptic curves. Applications of this work include
public key cryptography and coding theory, having direct impact
in e-commerce and telecommunications.
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Cohomology enhanced: an application of enriched and higher categories. Motivated by the needs of physicists, computer scientists, and
colleagues in other similar fields, mathematicians study highly
complicated structures which are typically hard to understand
completely in concrete terms. Cohomology is an invaluable technical
tool which allows data to be extracted from these complex structures.
This project will involve a radical expansion in scope of the amount
and type of data so extr ....Cohomology enhanced: an application of enriched and higher categories. Motivated by the needs of physicists, computer scientists, and
colleagues in other similar fields, mathematicians study highly
complicated structures which are typically hard to understand
completely in concrete terms. Cohomology is an invaluable technical
tool which allows data to be extracted from these complex structures.
This project will involve a radical expansion in scope of the amount
and type of data so extracted. This is made possible by the most
recent advances in higher-dimensional category theory.Read moreRead less
Invariants of higher-dimensional categories, with applications. Complex systems in mathematics are difficult to tell apart so one constructs simpler structures from them. These structures must be equal, isomorphic or equivalent when the original systems are equivalent; the word invariant is used for such constructions. Higher-dimensional categories are complex structures that are currently gaining a lot of attention from mathematicians, physicists and computer scientists because of developing ....Invariants of higher-dimensional categories, with applications. Complex systems in mathematics are difficult to tell apart so one constructs simpler structures from them. These structures must be equal, isomorphic or equivalent when the original systems are equivalent; the word invariant is used for such constructions. Higher-dimensional categories are complex structures that are currently gaining a lot of attention from mathematicians, physicists and computer scientists because of developing applications in those fields. This project will establish and study invariants for higher-dimensional categories which will be tested by examining their viability for producing results in group theory and homotopy theory.Read moreRead less
Applicable categorical structures. Mathematical research, like other endeavours, operates in specified environments: a space of numbers or vectors, a category of sets perhaps with extra structure, or a category of spaces. Often the environment is a specific category and analysis is internal to that. The novelty of category theory is that it applies also to external relations among the various environments. The direction of our work is motivated by aspects of mathematics, theoretical physics, and ....Applicable categorical structures. Mathematical research, like other endeavours, operates in specified environments: a space of numbers or vectors, a category of sets perhaps with extra structure, or a category of spaces. Often the environment is a specific category and analysis is internal to that. The novelty of category theory is that it applies also to external relations among the various environments. The direction of our work is motivated by aspects of mathematics, theoretical physics, and computer science. Such work underpins the capacity of the private sector by providing skilled graduates and enhancing the capabilities of the economy. Australia must maintain expertise in basic science and technology to be ready for uncertain future demands.
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HIGHER CATEGORICAL STRUCTURES IN HOMOTOPY THEORY AND HOMOLOGICAL ALGEBRA. This proposal falls in an area of research, that of higher categories, which has been receiving a lot of attention in recent years and which has applications to diverse areas of mathematics. The proposed research will contribute to continue the prominent role of Australian Research in this rapidly expanding field. History has proved that fundamental research in pure mathematics in the long term produces major and often une ....HIGHER CATEGORICAL STRUCTURES IN HOMOTOPY THEORY AND HOMOLOGICAL ALGEBRA. This proposal falls in an area of research, that of higher categories, which has been receiving a lot of attention in recent years and which has applications to diverse areas of mathematics. The proposed research will contribute to continue the prominent role of Australian Research in this rapidly expanding field. History has proved that fundamental research in pure mathematics in the long term produces major and often unexpected outcomes in applied sciences which have a direct impact on society. The area of higher categories has already proved to have an impact on applied fields such as computer science.Read moreRead less
Category theory arising from geometry, algebra, computer science and physics. Category theory is a branch of mathematics concerned with transformation and composition. It provides an algebra of wide-spread applicability for the synthesis of systems and processes in fields as diverse as geometry, physics and computer science, and also in mathematics itself. Often it can be used to clarify and simplify the learning, teaching and development of mathematics. The aim of this project is to develop the ....Category theory arising from geometry, algebra, computer science and physics. Category theory is a branch of mathematics concerned with transformation and composition. It provides an algebra of wide-spread applicability for the synthesis of systems and processes in fields as diverse as geometry, physics and computer science, and also in mathematics itself. Often it can be used to clarify and simplify the learning, teaching and development of mathematics. The aim of this project is to develop the general theory of categories and specifically to investigate aspects appropriate to algebra, physics and computer science.
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