Verifying the Riemann hypothesis to large heights: theory and applications. This project aims to verify the Riemann hypothesis to a record height and apply this verification to the distribution of prime numbers. The Riemann hypothesis (an open problem for 150 years) is ubiquitous in analytic number theory and prevalent in many other areas of mathematics. This project plans to use state-of-the-art computational hardware and the mathematical and algorithmic expertise of the investigators to verify ....Verifying the Riemann hypothesis to large heights: theory and applications. This project aims to verify the Riemann hypothesis to a record height and apply this verification to the distribution of prime numbers. The Riemann hypothesis (an open problem for 150 years) is ubiquitous in analytic number theory and prevalent in many other areas of mathematics. This project plans to use state-of-the-art computational hardware and the mathematical and algorithmic expertise of the investigators to verify the Riemann hypothesis several orders of magnitude further than what is currently known. A secondary aim is to apply this new verification to a multitude of results in analytic number theory: this would provide future researchers with vastly superior results.Read moreRead less
Explicit methods in number theory: Computation, theory and application. This project aims to use explicit estimates to unify three problems in number theory: primitive roots, Diophantine quintuples, and linear independence of zeroes of the Riemann zeta-function. It will use computational and analytic number theory to reduce the quintuples problem to a soluble level. Pursuing relations between the zeta zeroes will overhaul many current results. This project will apply its findings about primitive ....Explicit methods in number theory: Computation, theory and application. This project aims to use explicit estimates to unify three problems in number theory: primitive roots, Diophantine quintuples, and linear independence of zeroes of the Riemann zeta-function. It will use computational and analytic number theory to reduce the quintuples problem to a soluble level. Pursuing relations between the zeta zeroes will overhaul many current results. This project will apply its findings about primitive roots to signal processing, cryptography and cybersecurity.Read moreRead less
Modular Index Theory. This project capitilises on Australian advances in mathematics, particularly noncommutative geometry. It will maintain and extend Australia's prominence in this subject, providing excellent opportunities for young researchers via the research networks this project will establish. Being at the interface of ideas in mathematics and physics, there is potential for future technological spin offs for Australia.
Invariants for dynamics via operator algebras. Dynamics is the study of how the universe changes with time. At the quantum level, dynamics is highly unintuitive, and the sophisticated techniques of operator algebras are needed to describe it. This project will perfect new operator-algebraic tools to extract valuable new information about the behaviour of dynamical systems.
The boundaries of index theory. In recent years there has been an influx of new ideas from other disciplines into mathematics and this has led to major advances in many areas, notably geometry and topology. Classical problems have been solved and new perspectives exposed. In this spirit this project will use the methods of noncommutative analysis and noncommutative geometry to extend the mathematical area of spectral geometry. A primary objective is to determine how the geometric and differentia ....The boundaries of index theory. In recent years there has been an influx of new ideas from other disciplines into mathematics and this has led to major advances in many areas, notably geometry and topology. Classical problems have been solved and new perspectives exposed. In this spirit this project will use the methods of noncommutative analysis and noncommutative geometry to extend the mathematical area of spectral geometry. A primary objective is to determine how the geometric and differential structure of certain spaces interacts with the new spectral invariants that will be introduced. The project aims to obtain more subtle and refined information about these spaces. In this fashion it expects to resolve several long standing questions in mathematics. Read moreRead less
Physical realisation of enriched quantum symmetries. This project aims to investigate fundamental mathematical structures in modern category theory, providing an algebraic description of physical systems including topological order and conformal field theory. The project will study quantum symmetry, and classify and construct new classes of conformal field theories, using novel tools from enriched category theory, modular forms, and lattice gauge theory.
The main goal is to understand the lands ....Physical realisation of enriched quantum symmetries. This project aims to investigate fundamental mathematical structures in modern category theory, providing an algebraic description of physical systems including topological order and conformal field theory. The project will study quantum symmetry, and classify and construct new classes of conformal field theories, using novel tools from enriched category theory, modular forms, and lattice gauge theory.
The main goal is to understand the landscape of topological and conformal field theories, laying the foundation for new technologies based on topological order. This timely project capitalises on the recent arrival of subfactor experts in Australia, and builds capacity in mathematical research and international links in a cutting edge field.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE170101128
Funder
Australian Research Council
Funding Amount
$360,000.00
Summary
Homological methods in combinatorics, algebra and geometry. This project aims to solve problems in graph theory, lattice theory and geometry using algebraic techniques. The techniques and language provided by this algebraic approach will be used to gain fresh insight into classical problems, prove stronger theorems and uncover connections between different areas. This project intends to integrate Australia’s strength in homological algebra and category theory with applications in various differe ....Homological methods in combinatorics, algebra and geometry. This project aims to solve problems in graph theory, lattice theory and geometry using algebraic techniques. The techniques and language provided by this algebraic approach will be used to gain fresh insight into classical problems, prove stronger theorems and uncover connections between different areas. This project intends to integrate Australia’s strength in homological algebra and category theory with applications in various different fields of mathematics. This is expected to provide tools for further investigation of applications in other fields, including computer science and combinatorial optimisation.Read moreRead less
Symmetries of subfactors. A subfactor is a mathematical object that encodes "quantum" symmetries which may be thought of as generalisations of group symmetries. This project will study subfactors and classify families of subfactor symmetries which include the exotic subfactors of small index. It will also develop computational tools for analysing and cataloguing these symmetries. This project contributes to the development of operator algebra theory, and the new mathematical fields of quantum al ....Symmetries of subfactors. A subfactor is a mathematical object that encodes "quantum" symmetries which may be thought of as generalisations of group symmetries. This project will study subfactors and classify families of subfactor symmetries which include the exotic subfactors of small index. It will also develop computational tools for analysing and cataloguing these symmetries. This project contributes to the development of operator algebra theory, and the new mathematical fields of quantum algebra and quantum topology; it also has applications to physical models.Read moreRead less
A new theory of Aboriginal art. The paradigms developed by scholars have a huge impact on the value of art. After the Australian artworld repositioned Aboriginal art from the frame of 'primitive art' to that of contemporary art, the price of and market for Aboriginal art increased enormously. However, Aboriginal art failed to penetrate the international contemporary artworld because the paradigms of contemporary art within which it was framed were outmoded. Through examining the production of Ab ....A new theory of Aboriginal art. The paradigms developed by scholars have a huge impact on the value of art. After the Australian artworld repositioned Aboriginal art from the frame of 'primitive art' to that of contemporary art, the price of and market for Aboriginal art increased enormously. However, Aboriginal art failed to penetrate the international contemporary artworld because the paradigms of contemporary art within which it was framed were outmoded. Through examining the production of Aboriginal art from the perspectives of its producers and recent globalised contemporary art practices, this project aims to develop a new theory of Aboriginal art that is more aligned with the thinking of current curators who set the agenda at the upper end of the market.Read moreRead less
Banking System Competition and the Macro-economy. Australia has one of the most concentrated banking sectors in the world, generating concerns regarding its efficiency. This project aims to develop unified frameworks to understand and evaluate quantitatively how the structure of the banking industry affects the macro-economy and provide policy recommendations for establishing a healthy and efficient banking industry. This project expects to improve understanding of the welfare trade-off between ....Banking System Competition and the Macro-economy. Australia has one of the most concentrated banking sectors in the world, generating concerns regarding its efficiency. This project aims to develop unified frameworks to understand and evaluate quantitatively how the structure of the banking industry affects the macro-economy and provide policy recommendations for establishing a healthy and efficient banking industry. This project expects to improve understanding of the welfare trade-off between bank competition and economic well-being to enable policymakers to better determine the optimal concentration of banking sector in Australia. This will enhance the productivity and international competitiveness of Australia’s financial system and the broader economy.Read moreRead less