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
Stability conditions: their topology and applications. This project aims to answer questions about the topology of the space of stability conditions, which has emerged as a central object in a number of different mathematical areas in the past two decades. The proposed work will have important consequences in representation theory, group theory, and algebraic geometry. The project shows that tools from previously unrelated areas, including discontinous differential equations and discrete dynam ....Stability conditions: their topology and applications. This project aims to answer questions about the topology of the space of stability conditions, which has emerged as a central object in a number of different mathematical areas in the past two decades. The proposed work will have important consequences in representation theory, group theory, and algebraic geometry. The project shows that tools from previously unrelated areas, including discontinous differential equations and discrete dynamical systems, are crucial in the theory of stability conditions. Potential benefits include the resolution of outstanding conjectures in mathematics, the initiation of new connections between different areas of mathematics, and the introduction of machine learning techniques into mathematical research.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: DE240100447
Funder
Australian Research Council
Funding Amount
$438,847.00
Summary
The geometry of braids and triangulated categories. Triangulated categories play a central role in geometry, algebra, and topology. Their study can uncover deep structure connecting different areas of mathematics. This project aims to use novel approaches to answer fundamental questions about triangulated categories and their symmetries. These symmetries are encoded by braids, which are important objects with many applications across science. The project is expected to benefit Australia by stimu ....The geometry of braids and triangulated categories. Triangulated categories play a central role in geometry, algebra, and topology. Their study can uncover deep structure connecting different areas of mathematics. This project aims to use novel approaches to answer fundamental questions about triangulated categories and their symmetries. These symmetries are encoded by braids, which are important objects with many applications across science. The project is expected to benefit Australia by stimulating research in mathematics and computer science. It will invite connections with leading experts and students around the world and encourage overseas collaboration. There is a potential long-term benefit to cybersecurity, towards the development of new encryption schemes based on braids.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
Symmetry: Groups, Graphs, Number Fields and Loops. Exploiting symmetry can greatly simplify complex mathematical problems. This project aims to apply the powerful Classification of Finite Simple Groups to advance our understanding of the internal structure of number fields, highly symmetric graphs, and algebraic structures associated with Latin squares. The project expects to generate new constructions and classifications utilising group theory. Expected outcomes include resolutions of major ope ....Symmetry: Groups, Graphs, Number Fields and Loops. Exploiting symmetry can greatly simplify complex mathematical problems. This project aims to apply the powerful Classification of Finite Simple Groups to advance our understanding of the internal structure of number fields, highly symmetric graphs, and algebraic structures associated with Latin squares. The project expects to generate new constructions and classifications utilising group theory. Expected outcomes include resolutions of major open problems in each area as well as innovative methods for studying algebraic and combinatorial structures based on group actions. Expected benefits include enhanced international collaboration, and highly trained mathematicians to strengthen Australia’s research standing in fundamental science.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