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: DE140101366
Funder
Australian Research Council
Funding Amount
$376,527.00
Summary
Fully nonlinear partial differential equations in optimisation and applications. Fully nonlinear partial differential equations of Monge-Ampere type and their applications in optimal transportation have been studied intensively in the past two decades. Optimal transportation is a subject in linear optimisation. This project will develop a new theory on Monge-Ampere type equations in nonlinear optimisation, which is a much broader area with many significant applications. This project will investi ....Fully nonlinear partial differential equations in optimisation and applications. Fully nonlinear partial differential equations of Monge-Ampere type and their applications in optimal transportation have been studied intensively in the past two decades. Optimal transportation is a subject in linear optimisation. This project will develop a new theory on Monge-Ampere type equations in nonlinear optimisation, which is a much broader area with many significant applications. This project will investigate fundamental properties of solutions to this problem and explore further real-world applications. The outcomes of this project will have a substantial impact on partial differential equations and related research subjects. Read moreRead less
Monge-Ampere equations and applications. The Monge-Ampere equation is a premier fully nonlinear partial differential equation with significant applications in geometry, physics and applied science. Building upon breakthroughs made by the proposers in previous grant research, this project aims to resolve challenging problems involving Monge-Ampere type equations and applications. The project goal is to establish new regularity theory and classify singularity profile for solutions to Monge-Ampere ....Monge-Ampere equations and applications. The Monge-Ampere equation is a premier fully nonlinear partial differential equation with significant applications in geometry, physics and applied science. Building upon breakthroughs made by the proposers in previous grant research, this project aims to resolve challenging problems involving Monge-Ampere type equations and applications. The project goal is to establish new regularity theory and classify singularity profile for solutions to Monge-Ampere type equation arising in applied sciences, by introducing new ideas and developing innovative cutting-edge techniques. Expected outcomes include resolution of outstanding open problems and continuing enhancement of Australian leadership and expertise in a major area of mathematics.
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Number Theoretic Methods in Cryptography. It is well known that Number Theory, besides its intrinsic beauty, provides many powerful tools for modern Cryptography. The aim of the project is to formulate and solve new and important mathematical problems, which lie in the background of modern cryptography. They are also of independent value for pure mathematics because they very often stimulate new approaches to and new surprising points of view on classical results and methods. The main outcome w ....Number Theoretic Methods in Cryptography. It is well known that Number Theory, besides its intrinsic beauty, provides many powerful tools for modern Cryptography. The aim of the project is to formulate and solve new and important mathematical problems, which lie in the background of modern cryptography. They are also of independent value for pure mathematics because they very often stimulate new approaches to and new surprising points of view on classical results and methods. The main outcome will be advancing our theoretical knowledge about several major cryptosystems. The project will extend and enrich the area of applications of mathematics to cryptography and related areas.Read moreRead less
Mathematics of Cryptography. The Australian economy and society requires fast, reliable, and secure communication. Current first-generation security solutions are not capable of supporting the efficiency and scalability requirements of mass-market adoption of wireless and embedded consumer applications. New security infrastructures are emerging and must be carefully, but rapidly, defined. Thus developing new mathematically solid tools in this area is an important and urgent tasks. In addition, t ....Mathematics of Cryptography. The Australian economy and society requires fast, reliable, and secure communication. Current first-generation security solutions are not capable of supporting the efficiency and scalability requirements of mass-market adoption of wireless and embedded consumer applications. New security infrastructures are emerging and must be carefully, but rapidly, defined. Thus developing new mathematically solid tools in this area is an important and urgent tasks. In addition, the intended work advances our knowledge of the theory and the quality of our culture. As such, it will promote the Australian science and will also have many practical applications in Cryptography, Computer Security and E-Commerce.Read moreRead less
Mathematics of Cryptography. The Australian society and economy requires fast, reliable, and secure communication. First-generation security solutions are not capable of supporting the efficiency and scalability requirements of mass-market adoption of wireless and embedded consumer applications. New security infrastructures are emerging and must be carefully, but rapidly, defined. Thus developing new mathematically solid tools in this area is one of the most important and urgent tasks. Besides, ....Mathematics of Cryptography. The Australian society and economy requires fast, reliable, and secure communication. First-generation security solutions are not capable of supporting the efficiency and scalability requirements of mass-market adoption of wireless and embedded consumer applications. New security infrastructures are emerging and must be carefully, but rapidly, defined. Thus developing new mathematically solid tools in this area is one of the most important and urgent tasks. Besides, the intended work advances our knowledge of the theory and the quality of our culture. As such, it will promote the Australian science and will also have many practical applications in Cryptography, Computer Security and E-Commerce.Read moreRead less
Analysis of nonlinear partial differential equations describing singular phenomena. This project will advance knowledge on a huge variety of systems with applications across the sciences by providing new methods to investigate nonlinear partial differential equations with singularities. The analysis of many models describing physical and biological systems relies on such equations.
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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New constructions and techniques for tensor categories. The goal of this project is to make fundamental advances in the structure theory of tensor categories. Such categories play crucial roles in numerous fields of mathematics, physics and beyond. New methods, theory and examples will be developed, inspired by algebra, representation theory and geometry. These will then be applied in the foundational study of tensor categories for (dis)proving several of the most important open conjectures in t ....New constructions and techniques for tensor categories. The goal of this project is to make fundamental advances in the structure theory of tensor categories. Such categories play crucial roles in numerous fields of mathematics, physics and beyond. New methods, theory and examples will be developed, inspired by algebra, representation theory and geometry. These will then be applied in the foundational study of tensor categories for (dis)proving several of the most important open conjectures in the field. This will open new perspectives for applications in other areas, most notably in representation theory. Other benefits include enhanced international collaboration and scientific capacity in Australia.Read moreRead less
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