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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Pictures for Operator Algebras: higher rank graphs. The runaway success of graph C*-algebras pioneered at the University of Newcastle places Australia at the forefront of an exciting new research field. Higher-dimensional graphs and their algebras are a promising new direction, and this project will keep Australian researchers at its cutting edge.
This project will involve and train talented young researchers who will contribute to the Mathematical outcomes of the current project. Their invol ....Pictures for Operator Algebras: higher rank graphs. The runaway success of graph C*-algebras pioneered at the University of Newcastle places Australia at the forefront of an exciting new research field. Higher-dimensional graphs and their algebras are a promising new direction, and this project will keep Australian researchers at its cutting edge.
This project will involve and train talented young researchers who will contribute to the Mathematical outcomes of the current project. Their involvement will train them in the techniques of modern Mathematics and furnish them with important international research connections. In doing so we shall be laying a strong foundation for the future of Australian Mathematical research.Read moreRead less
Foundations of higher dimensional homological algebra. Recent discoveries in physics and mathematics led to the understanding that classical mathematics is only 'the tip of the iceberg' of the higher-dimensional structures that are ultimately behind the laws of Nature. Australia has always been in the forefront of research in Category Theory, and due to that position, has a unique opportunity to participate in the early stages of developments of Higher Category Theory and Higher Dimensiona ....Foundations of higher dimensional homological algebra. Recent discoveries in physics and mathematics led to the understanding that classical mathematics is only 'the tip of the iceberg' of the higher-dimensional structures that are ultimately behind the laws of Nature. Australia has always been in the forefront of research in Category Theory, and due to that position, has a unique opportunity to participate in the early stages of developments of Higher Category Theory and Higher Dimensional Homological Algebra. This will allow Australia to be in the forefront of the subsequent technological development and to reap the economical, social and intellectual benefits related to it.
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Categorical structures in string theory. The proposal is a contribution to the mathematics of fundamental laws of nature. Developments in string theory are unfolding internationally from top physicists and mathematicians. Basic research by our expert group of category theorists will reach out into the Australian community to varying degrees through our own teaching, vacation scholars, media interviews, and links with our academic colleagues in other disciplines. Such basic research underpins ....Categorical structures in string theory. The proposal is a contribution to the mathematics of fundamental laws of nature. Developments in string theory are unfolding internationally from top physicists and mathematicians. Basic research by our expert group of category theorists will reach out into the Australian community to varying degrees through our own teaching, vacation scholars, media interviews, and links with our academic colleagues in other disciplines. Such basic research 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.Read moreRead less
Noncommutative geometry of groups acting on buildings. Consider a tiling of the plane by triangles, where each triangle is labeled by an element of a finite alphabet. Suppose that only certain pairs of labels are allowed to be adjacent to each other in each direction. The tiled planes can be pasted together to form the abstract mathematical object known as a building. This building and its boundary, give rise to new families of C*-algebras and groups. The algebras have a rich structure which it ....Noncommutative geometry of groups acting on buildings. Consider a tiling of the plane by triangles, where each triangle is labeled by an element of a finite alphabet. Suppose that only certain pairs of labels are allowed to be adjacent to each other in each direction. The tiled planes can be pasted together to form the abstract mathematical object known as a building. This building and its boundary, give rise to new families of C*-algebras and groups. The algebras have a rich structure which it is proposed to investigate and link with geometric properties of the groups. New insights into geometry, dynamics and algebra are expected.Read moreRead less
Functorial operadic calculus. Further progress in the foundations of quantum physics, algebra and geometry requires a development of mathematical theories governed by the complicated algebra of higher-dimensional substitutions. The study of this algebra is the main focus of this project. It will allow Australia to remain at the forefront of research into the fundamental laws of Nature and subsequent technological development and to reap the economic, social and intellectual benefits relate ....Functorial operadic calculus. Further progress in the foundations of quantum physics, algebra and geometry requires a development of mathematical theories governed by the complicated algebra of higher-dimensional substitutions. The study of this algebra is the main focus of this project. It will allow Australia to remain at the forefront of research into the fundamental laws of Nature and subsequent technological development and to reap the economic, social and intellectual benefits related to this developmentRead moreRead less