Categorical representation theory and applications. Symmetry is everywhere, and nature is designed symmetrically: Snails make their shells, spiders design their webs, and bees build hexagonal honeycombs, all based on the concept of symmetry. Symmetry is a general principle which plays an important role in various areas of knowledge and perception, ranging from arts to natural sciences and mathematics.
The 21th century way of the study of symmetries is categorical representation theory. The proj ....Categorical representation theory and applications. Symmetry is everywhere, and nature is designed symmetrically: Snails make their shells, spiders design their webs, and bees build hexagonal honeycombs, all based on the concept of symmetry. Symmetry is a general principle which plays an important role in various areas of knowledge and perception, ranging from arts to natural sciences and mathematics.
The 21th century way of the study of symmetries is categorical representation theory. The project aims are to strengthen this young field by advancing the theory and by finding applications from where its significance arises. The outcome will be new results on categorical representations and this will have benefits within mathematics, cryptography and also in physics/chemistry in the long run.Read moreRead less
Braid groups via representation theory and machine learning. This project aims to address questions about the representation theory of braid groups with important consequences in low-dimensional topology. This project expects to make significant progress on central open problems surrounding knot invariants, and create new tools that will have wide applicability in representation theory. It will pioneer the use of highly innovative methods from category theory and machine learning recently develo ....Braid groups via representation theory and machine learning. This project aims to address questions about the representation theory of braid groups with important consequences in low-dimensional topology. This project expects to make significant progress on central open problems surrounding knot invariants, and create new tools that will have wide applicability in representation theory. It will pioneer the use of highly innovative methods from category theory and machine learning recently developed by the investigators. Potential benefits of this project include: the resolution of important long-standing conjectures about braid groups, the development of emerging technology with significant implications for representation theory, and the training of Australian scientists in a vital area of research.Read moreRead less
Categorical geometry and perfect group schemes. The aims of this project are to construct novel geometric theories based on newly discovered tensor categories, to apply the theories to solve open problems in representation theory, algebra and category theory, and to establish profitable new connections between the influential theories of affine group schemes and classifying spaces. The geometric theories will be developed in a universal way, generalising both classical algebraic geometry and sup ....Categorical geometry and perfect group schemes. The aims of this project are to construct novel geometric theories based on newly discovered tensor categories, to apply the theories to solve open problems in representation theory, algebra and category theory, and to establish profitable new connections between the influential theories of affine group schemes and classifying spaces. The geometric theories will be developed in a universal way, generalising both classical algebraic geometry and super geometry from physics, and specialising to infinitely many new theories. This universality ensures a significantly broader basis for long term applications of geometry in many areas of science. Other benefits include enhanced international collaboration and scientific capacity in Australia.Read moreRead less
Graded Symmetry in Algebra and Analysis. This project will study graded symmetries in mathematics by modelling them as groupoids and inverse semigroups. Groupoids have been at the centre of mathematical interest for a long time, but have gained special prominence in recent years as a focal point for algebra, analysis and dynamics. The majority of groupoids can be naturally graded. The project introduces graded combinatorial invariants for groupoids (such as graded homology) and relates them to t ....Graded Symmetry in Algebra and Analysis. This project will study graded symmetries in mathematics by modelling them as groupoids and inverse semigroups. Groupoids have been at the centre of mathematical interest for a long time, but have gained special prominence in recent years as a focal point for algebra, analysis and dynamics. The majority of groupoids can be naturally graded. The project introduces graded combinatorial invariants for groupoids (such as graded homology) and relates them to their Steinberg and C*-algebra counterparts (such as graded K-theory). The outcome is to give sought-after unified invariants bridging algebra and analysis, and to exhaust the class of groupoids for which these much richer invariants will furnish a complete classification. Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE230100303
Funder
Australian Research Council
Funding Amount
$352,200.00
Summary
New Foundations for Algebraic Geometry. Differential calculus is one of the most important and widely applied areas of mathematics. Differential categories are a modern foundational theory of differential calculus with applications throughout mathematics and computing. This project aims to use differential categories to create new foundations for algebraic geometry, and to generate new knowledge on the connection between algebraic and differential geometry. The generality of these foundations wi ....New Foundations for Algebraic Geometry. Differential calculus is one of the most important and widely applied areas of mathematics. Differential categories are a modern foundational theory of differential calculus with applications throughout mathematics and computing. This project aims to use differential categories to create new foundations for algebraic geometry, and to generate new knowledge on the connection between algebraic and differential geometry. The generality of these foundations will allow for novel applications of algebraic geometry with significant benefits to computer science, such as in machine learning and differentiable programming. We expect this to build Australia's profile in these important fields and help train the next generation of mathematicians.Read moreRead less
Australian Laureate Fellowships - Grant ID: FL230100256
Funder
Australian Research Council
Funding Amount
$3,359,669.00
Summary
Unlocking the secrets of modular representations. This Fellowship aims to greatly increase our understanding of the fundamental symmetries of discrete structures, like those present in computer science and cryptography. The research will generate transformative new knowledge in pure mathematics concerning the representations of finite groups, problems that have been unsolved for over a century. Expected outcomes of this fellowship include new algorithms to compute far beyond what is currently p ....Unlocking the secrets of modular representations. This Fellowship aims to greatly increase our understanding of the fundamental symmetries of discrete structures, like those present in computer science and cryptography. The research will generate transformative new knowledge in pure mathematics concerning the representations of finite groups, problems that have been unsolved for over a century. Expected outcomes of this fellowship include new algorithms to compute far beyond what is currently possible and a new understanding of the arithmetic difficulties present. Key benefits will be seen in the development of an emerging technology with significant implications for mathematics, and the training of Australian scientists in sophisticated theory and large-scale computation in concert.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
Class numbers and discriminants: algebraic and analytic number theory meet. This project aims to investigate connections between analytic and algebraic number theory utilising the theoretical and computational expertise of the research group in number theory at UNSW Canberra. The potential findings are highly significant since the innovative generation of new fundamental knowledge will expand the field, and have cryptographic applications.
The expected outcomes include increased capacity in fun ....Class numbers and discriminants: algebraic and analytic number theory meet. This project aims to investigate connections between analytic and algebraic number theory utilising the theoretical and computational expertise of the research group in number theory at UNSW Canberra. The potential findings are highly significant since the innovative generation of new fundamental knowledge will expand the field, and have cryptographic applications.
The expected outcomes include increased capacity in fundamental science and greater understanding of classical and quantum cryptographic protocols. This project will provide the additional, and substantial, benefit of generating research output, training HDR students, and contributions towards national security.Read moreRead less
Ubiquity of Kloosterman sums in Number Theory and Beyond. This project aims to seek new methods of investigating Kloosterman sums by
combining an algebraic geometry approach with an analytic approach to develop one
powerful, unified method. Its significance lies in expected pivotal advances towards
several fundamental problems which lie at the heart of number theory such as
the Dirichlet Divisor Problem and asymptotic formulas for moments of L-functions.
The expected outcome of the project is ....Ubiquity of Kloosterman sums in Number Theory and Beyond. This project aims to seek new methods of investigating Kloosterman sums by
combining an algebraic geometry approach with an analytic approach to develop one
powerful, unified method. Its significance lies in expected pivotal advances towards
several fundamental problems which lie at the heart of number theory such as
the Dirichlet Divisor Problem and asymptotic formulas for moments of L-functions.
The expected outcome of the project is to provide a deeper understanding of the
intriguing nature of Kloosterman sums and thus open new perspectives for
applications in analytic number theory. This will provide
substantial benefits for other areas such as cryptography by deepening our understanding of pseudorandom sequences.Read moreRead less
Interplay between Ergodic Theory, Additive Combinatorics and Ramsey Theory. This project aims to address fundamental problems in Number Theory and Combinatorics by developing new innovative ergodic theoretic methods. Expected outcomes of the project include finding new patterns in dense subsets of trees, obtaining rigorous number-theoretic results emphasising the independence of addition and multiplication, finding infinite patterns in dense subsets of primes, and developing a multi-dimensional ....Interplay between Ergodic Theory, Additive Combinatorics and Ramsey Theory. This project aims to address fundamental problems in Number Theory and Combinatorics by developing new innovative ergodic theoretic methods. Expected outcomes of the project include finding new patterns in dense subsets of trees, obtaining rigorous number-theoretic results emphasising the independence of addition and multiplication, finding infinite patterns in dense subsets of primes, and developing a multi-dimensional analogue of the dense model theory for primes. This project will provide significant benefits to Australian research via an intensive collaboration with best international and Australian researchers working in ergodic and number theory as well as will be used to educate a new generation of Australian students. Read moreRead less