Hypergraph models for complex discrete systems. This project aims to better understand the structure and properties of very large hypergraphs of various kinds. Hypergraphs are very general mathematical objects which can be used to model complex discrete systems. They arise naturally in many areas such as ecology, chemistry and computer science. Despite this, our theoretical understanding of very large, or random, hypergraphs lags far behind the intensely-studied special case of graphs. This proj ....Hypergraph models for complex discrete systems. This project aims to better understand the structure and properties of very large hypergraphs of various kinds. Hypergraphs are very general mathematical objects which can be used to model complex discrete systems. They arise naturally in many areas such as ecology, chemistry and computer science. Despite this, our theoretical understanding of very large, or random, hypergraphs lags far behind the intensely-studied special case of graphs. This project will answer many fundamental questions about large, random hypergraphs. The expected outcomes of the project also include new tools for working with hypergraphs, such as efficient algorithms for sampling hypergraphs. These outcomes will benefit researchers who use hypergraphs in their work and will enhance Australia's reputation for research in this area.Read moreRead less
Enumeration and properties of large discrete structures. This project aims to study a fundamental property of random graphs, by further developing a recently introduced approach to the problem of enumerating graphs with given degrees. Using this new method, the project expects to generate new knowledge on the number of connections that each node has with other nodes in a random graph, and to develop new strategies for counting the graphs or networks with a given property. The project expects to ....Enumeration and properties of large discrete structures. This project aims to study a fundamental property of random graphs, by further developing a recently introduced approach to the problem of enumerating graphs with given degrees. Using this new method, the project expects to generate new knowledge on the number of connections that each node has with other nodes in a random graph, and to develop new strategies for counting the graphs or networks with a given property. The project expects to produce new theoretical results as well as enhanced capabilities of mathematical research. Potential benefits arise through the uses of these theoretical combinatorial objects to study naturally occurring networks such as social networks, the network of the world wide web, and chemical compounds.Read moreRead less
Enumeration and random generation of contingency tables with given margins. This project aims to find algorithms to construct random tables of numbers having given totals across the rows and down the columns. The aim is also to study properties of such tables. A significant aspect of the project is that it is expected to cover scenarios where all existing methods fail, by deploying recently developed powerful techniques used for random networks in combinatorics. Expected outcomes of this project ....Enumeration and random generation of contingency tables with given margins. This project aims to find algorithms to construct random tables of numbers having given totals across the rows and down the columns. The aim is also to study properties of such tables. A significant aspect of the project is that it is expected to cover scenarios where all existing methods fail, by deploying recently developed powerful techniques used for random networks in combinatorics. Expected outcomes of this project include the development of efficient algorithms that can be used in statistics for identifying relationships between variables in large data sets. This would help bring Australia to the forefront of research in an area that is significant both in data analysis and in discrete mathematics.
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Triangulations: linking geometry and topology with combinatorics. Triangulations are the method of choice to represent geometric objects given by a finite sample of points. Prominent examples include the pictures produced by the finite element method, polytopes in optimisation, or surfaces in computer graphics.
Knowledge about the triangulations of an object and how they relate to each other is essential for these applications. Seemingly canonical and straightforward methods perform well - or n ....Triangulations: linking geometry and topology with combinatorics. Triangulations are the method of choice to represent geometric objects given by a finite sample of points. Prominent examples include the pictures produced by the finite element method, polytopes in optimisation, or surfaces in computer graphics.
Knowledge about the triangulations of an object and how they relate to each other is essential for these applications. Seemingly canonical and straightforward methods perform well - or not at all, depending on intricate and highly involved mathematical properties.
In this project we combine geometric and topological viewpoints to tackle high-profile questions about triangulations. This will unlock the full potential of combinatorial methods and practical algorithms in applications.Read moreRead less
Australian Laureate Fellowships - Grant ID: FL210100110
Funder
Australian Research Council
Funding Amount
$3,021,288.00
Summary
New Approaches to Understand How Form and Function Shape Complex Systems. As biology and medicine transform into quantitative sciences, existing mathematical methods are often inadequate to explain the data they generate. This project aims to unlock the potential of such biomedical data through the development of new mathematical approaches that combine concepts from pure and applied mathematics, statistics and data science, and then to investigate their ability to generate mechanistic insight i ....New Approaches to Understand How Form and Function Shape Complex Systems. As biology and medicine transform into quantitative sciences, existing mathematical methods are often inadequate to explain the data they generate. This project aims to unlock the potential of such biomedical data through the development of new mathematical approaches that combine concepts from pure and applied mathematics, statistics and data science, and then to investigate their ability to generate mechanistic insight into fundamental biomedical processes. In this way, the project expects to affect a paradigm shift in mathematical biology while strengthening Australia’s reputation as a world-leader in mathematical biology. An outcome from this project could be new mathematical models that guide decision making in the clinic.Read moreRead less
Diagram categories and transformation semigroups. A structural understanding of diagram categories is essential in many branches of mathematics and science. Despite this, very few methods for studying such categories are available, a fact this pure mathematics project seeks to rectify. By building strong bridges between diagram categories and semigroup theory, a field of abstract algebra that models transformation and change, the structure of diagram categories may be unlocked with powerful semi ....Diagram categories and transformation semigroups. A structural understanding of diagram categories is essential in many branches of mathematics and science. Despite this, very few methods for studying such categories are available, a fact this pure mathematics project seeks to rectify. By building strong bridges between diagram categories and semigroup theory, a field of abstract algebra that models transformation and change, the structure of diagram categories may be unlocked with powerful semigroup tools developed by the applicant investigator. Diagrammatic insights will also yield new ways to study semigroups, and the many other mathematical structures they interact with. Outcomes will have a lasting impact on both theories as well as the many fields influenced by them.Read moreRead less
New directions in Hecke algebras. To goal of this project is to make fundamental advances in representation theory, a powerful branch of mathematics focused on taking abstract mathematical structures and ``representing'' them in a concrete and useful way. In particular we aim to prove a series of long standing and influential conjectures by George Lusztig concerning the representation theory of Hecke algebras, objects which are ubiquitous in modern algebra. Our work will lead to new discoveries, ....New directions in Hecke algebras. To goal of this project is to make fundamental advances in representation theory, a powerful branch of mathematics focused on taking abstract mathematical structures and ``representing'' them in a concrete and useful way. In particular we aim to prove a series of long standing and influential conjectures by George Lusztig concerning the representation theory of Hecke algebras, objects which are ubiquitous in modern algebra. Our work will lead to new discoveries, a fundamentally deeper understanding of Kazhdan-Lusztig theory, and will drive future research. Benefits include enhanced international collaboration and increasing capacity in pure mathematics, especially in the cutting-edge area of representation theory.Read moreRead less
Categorification and KLR algebras. AIMS This project will solve three problems at the forefront of representation theory: the centre conjecture for graded Hecke algebras, concretely connecting crystals with KLR algebras and describing the grading and radical filtrations Specht modules.
SIGNIFICANCE Solving any of these problems will represent a serious advance in the field and have a lasting impact and creating new areas of research.
EXPECTED OUTCOMES We will remove major bottlenecks in our u ....Categorification and KLR algebras. AIMS This project will solve three problems at the forefront of representation theory: the centre conjecture for graded Hecke algebras, concretely connecting crystals with KLR algebras and describing the grading and radical filtrations Specht modules.
SIGNIFICANCE Solving any of these problems will represent a serious advance in the field and have a lasting impact and creating new areas of research.
EXPECTED OUTCOMES We will remove major bottlenecks in our understanding of KLR algebras.
BENEFITS In addition to the mathematical benefits, the skills and expertise that are required for, and will be enhanced by, this project are readily transferable and highly sought after by industry, including the financial, IT and education sectors.Read moreRead less
Affine flags, folded galleries and euclidean reflection groups. This project aims to answer important geometric questions about subvarieties of the affine flag variety which are fundamental to algebraic geometry and number theory. It will answer basic questions about these central objects of mathematics, affine flags and their subspaces, using powerful new methods which combine ideas from geometry and algebra. The project expects to include finding the patterns of non-emptiness of these subvar ....Affine flags, folded galleries and euclidean reflection groups. This project aims to answer important geometric questions about subvarieties of the affine flag variety which are fundamental to algebraic geometry and number theory. It will answer basic questions about these central objects of mathematics, affine flags and their subspaces, using powerful new methods which combine ideas from geometry and algebra. The project expects to include finding the patterns of non-emptiness of these subvarieties and formulae for their dimension. It will develop and apply new methods which combine folded galleries and the geometry of Euclidean reflection groups, and these methods will have applications in algebraic combinatorics and representation theory. The project will also inspire productive connections between geometric group theory, a new and fast-growing area, and the classical fields of algebraic geometry, algebraic combinatorics and representation theory.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE150101415
Funder
Australian Research Council
Funding Amount
$360,000.00
Summary
Higher Representation Theory. Representation theory is a field of mathematics with applications across the breadth of mathematical study in fields as diverse as number theory and physics. The study of higher (or categorical) representation theory is a modern set of tools that provides new insights into representation theoretic phenomena. This project aims to study categorified quantum groups and, in particular, the categorifications provided by diagrammatic algebras. The project aims to further ....Higher Representation Theory. Representation theory is a field of mathematics with applications across the breadth of mathematical study in fields as diverse as number theory and physics. The study of higher (or categorical) representation theory is a modern set of tools that provides new insights into representation theoretic phenomena. This project aims to study categorified quantum groups and, in particular, the categorifications provided by diagrammatic algebras. The project aims to further develop the theory of Khovanov-Lauda-Rouquier (KLR) algebras, providing important foundational results for future research to build upon.Read moreRead less