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
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: 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
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
Discovery Early Career Researcher Award - Grant ID: DE230100954
Funder
Australian Research Council
Funding Amount
$354,968.00
Summary
Partial Differential Equations, geometric aspects and applications. The study of Partial Differential Equations (PDEs) is a classical and prolific field of research having a fundamental role in the development of mathematical analysis and motivated by important applications in natural and applied sciences.
This project aims to obtain substantial progress in the field of PDEs. The area of mathematical research covered is extremely broad, at the confluence of analysis and geometry, and with many a ....Partial Differential Equations, geometric aspects and applications. The study of Partial Differential Equations (PDEs) is a classical and prolific field of research having a fundamental role in the development of mathematical analysis and motivated by important applications in natural and applied sciences.
This project aims to obtain substantial progress in the field of PDEs. The area of mathematical research covered is extremely broad, at the confluence of analysis and geometry, and with many applications to other areas of mathematics and natural and applied sciences. The results that will be obtained will produce a significant amount of new knowledge in this extremely difficult, but rapidly growing, field, by exploiting international scientific collaborations and interdisciplinary methods.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