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
Random Structures and Asymptotics. Discrete random structures have many uses in algorithms in computer science (for instance, random networks modelling computer link-ups), biology (for instance, random sequences modelling DNA) and engineering. New techniques for studying these structures will lead to powerful new results on their properties. The emphasis will be on the behaviour of the random structures when their size becomes large. With the advent of
more powerful computing techniques, it is ....Random Structures and Asymptotics. Discrete random structures have many uses in algorithms in computer science (for instance, random networks modelling computer link-ups), biology (for instance, random sequences modelling DNA) and engineering. New techniques for studying these structures will lead to powerful new results on their properties. The emphasis will be on the behaviour of the random structures when their size becomes large. With the advent of
more powerful computing techniques, it is often the large-scale behaviour which has relevance to the more diffucult computations being undertaken. The results are also of potential application to other areas of mathematics.Read moreRead less
Asymptotic Geometric Analysis and Machine Learning. Phenomena in large dimensions appear in a number of domains of Mathematics and adjacent domains of science (e.g. Computer Science), dealing with functions of infinitely growing number of parameters. Here, we focus on several questions naturally linked to Asymptotic Geometric Analysis which have natural applications to Statistical Learning Theory. We intend to use geometric, probabilistic and combinatorial methods to investigate these problems, ....Asymptotic Geometric Analysis and Machine Learning. Phenomena in large dimensions appear in a number of domains of Mathematics and adjacent domains of science (e.g. Computer Science), dealing with functions of infinitely growing number of parameters. Here, we focus on several questions naturally linked to Asymptotic Geometric Analysis which have natural applications to Statistical Learning Theory. We intend to use geometric, probabilistic and combinatorial methods to investigate these problems, with an emphasis on modern tools in Empirical Processes Theory and the theory of Random Matrices.Read moreRead less
A new model for random discrete structures: distributions, counting and sampling. Random discrete structures are used in countless applications across science for modelling complex systems. This project will study a new, very general model of random discrete structures which encapsulates both random networks and random matrices. This project will develop general tools for working with this model, thereby unlocking the model for use by practitioners in areas such as physics, biology, statistics a ....A new model for random discrete structures: distributions, counting and sampling. Random discrete structures are used in countless applications across science for modelling complex systems. This project will study a new, very general model of random discrete structures which encapsulates both random networks and random matrices. This project will develop general tools for working with this model, thereby unlocking the model for use by practitioners in areas such as physics, biology, statistics and cryptography. The questions that will be tackled are fundamental problems in probability, and include as special cases the analysis of subgraph distribution in models of random networks, and the joint distribution of entries of contingency tables, which are important in statistics.Read moreRead less
Australian Laureate Fellowships - Grant ID: FL120100125
Funder
Australian Research Council
Funding Amount
$1,796,966.00
Summary
Advances in the analysis of random structures and their applications. This project will provide new approaches, insights and results for probabilistic combinatorics. This area has contributed in exciting ways elsewhere in mathematics and provides versatile tools of widespread use in algorithmic computer science, with other applications in physics, coding theory for communications, and genetics.
Algebraic K-theory and groups. This project will study the K-theory of division algebras, their generalisation to Azumaya algebras and the nonstable K-theory of rings. Expected outcomes would enhance our understanding on the structure of these K groups.
The goal is to settle some of the most significant conjectures in the subject: Bak's solvability of nonstable K groups over rings and the Merkurjev-Suslin conjectures on reduced K theory of division rings.
The study of these problems contribu ....Algebraic K-theory and groups. This project will study the K-theory of division algebras, their generalisation to Azumaya algebras and the nonstable K-theory of rings. Expected outcomes would enhance our understanding on the structure of these K groups.
The goal is to settle some of the most significant conjectures in the subject: Bak's solvability of nonstable K groups over rings and the Merkurjev-Suslin conjectures on reduced K theory of division rings.
The study of these problems contributes to and draws from such topics as group theory, commutative ring theory, algebraic number theory and algebraic geometry.
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Homotopy theory: interactions with representation theory and moduli spaces. This proposal will involve young researchers and train them for problem solving in many fields, including management, the sciences, the financial industries, and the development of technologies. Furthermore, many of the projects in this proposal are collaborative and interdisciplinary. It is the CI's sincere hope that this proposal can help bolster communication amongst the wealth of topology, number theory, and mathe ....Homotopy theory: interactions with representation theory and moduli spaces. This proposal will involve young researchers and train them for problem solving in many fields, including management, the sciences, the financial industries, and the development of technologies. Furthermore, many of the projects in this proposal are collaborative and interdisciplinary. It is the CI's sincere hope that this proposal can help bolster communication amongst the wealth of topology, number theory, and mathematical physics experts in Australia. The research in these exciting areas of mathematics will contribute to maintaining Australia's position as a research leader in pure mathematics.
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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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Discovery Early Career Researcher Award - Grant ID: DE120102369
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Higher representation theory. Representation theory lies at the very centre of mathematics, with applications in all areas of mathematics and mathematical physics; at some level it is about observing the symmetries of a system and exploiting them, and this has been invaluable. This project will explore the forefront of the modern, higher version of this research field.
Discovery Early Career Researcher Award - Grant ID: DE150101161
Funder
Australian Research Council
Funding Amount
$330,000.00
Summary
Quantum Groups and Categorification in Geometric Representation Theory. Representation theory, the mathematical study of symmetry, has applications in diverse areas such as particle physics, computer science, and molecular biology. This project aims to use a new family of quantum groups to prove a network of conjectures about categorical and geometric representation theory. The project aims to answer a long-standing open question in geometric representation theory concerning two families of geom ....Quantum Groups and Categorification in Geometric Representation Theory. Representation theory, the mathematical study of symmetry, has applications in diverse areas such as particle physics, computer science, and molecular biology. This project aims to use a new family of quantum groups to prove a network of conjectures about categorical and geometric representation theory. The project aims to answer a long-standing open question in geometric representation theory concerning two families of geometric spaces underlying the theory of Lie groups. Representation theory is a strength of Australian mathematics, and this project aims to undertake pressing research at the forefront of this dynamic field.Read moreRead less