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: DE190101099
Funder
Australian Research Council
Funding Amount
$420,256.00
Summary
Representation theory: studies of symmetry shadows. This project aims to solve fundamental problems in representation theory by combining cutting-edge techniques and developing novel higher level structures. Representation theory is the mathematical study of symmetry, an essential concept in science. Since the 1990s, mathematicians have been observing shadows of a more general notion of symmetry but so far have failed to explain it. Expected outcomes include a structural explanation of these sh ....Representation theory: studies of symmetry shadows. This project aims to solve fundamental problems in representation theory by combining cutting-edge techniques and developing novel higher level structures. Representation theory is the mathematical study of symmetry, an essential concept in science. Since the 1990s, mathematicians have been observing shadows of a more general notion of symmetry but so far have failed to explain it. Expected outcomes include a structural explanation of these shadows, new mathematical software to understand them and solutions to important conjectures. This project will make a significant contribution to the field of representation theory, with ramifications in mathematical physics and computer science.Read moreRead less
Representations of arithmetic groups and their associated zeta functions. This project aims to investigate deep connections between number theory and group theory by studying linear actions of arithmetic groups. Arithmetic groups are used in geometry, dynamics, number theory and other areas of pure mathematics. This project will study their representations from two perspectives. First, it will establish properties of the associated zeta functions to resolve open problems about the asymptotic beh ....Representations of arithmetic groups and their associated zeta functions. This project aims to investigate deep connections between number theory and group theory by studying linear actions of arithmetic groups. Arithmetic groups are used in geometry, dynamics, number theory and other areas of pure mathematics. This project will study their representations from two perspectives. First, it will establish properties of the associated zeta functions to resolve open problems about the asymptotic behaviour of the dimensions of the irreducible representations. Second, it will explore the evolution of representations across families of groups under new induction and restriction functors, in analogy with creation and annihilation operators in physics. The project will enhance Australia's capacity in representation theory and group theory, the mathematics that underline symmetry in nature.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE190101231
Funder
Australian Research Council
Funding Amount
$390,000.00
Summary
Integrable systems from gauge theories. This project aims to construct and describe three important integrable systems and to produce new significant results in representation theory and mathematical physics. The focus of the project is the theory of integrable systems, a breakthrough area in modern mathematics, and it will use tools from geometric representation theory to study the interrelation between geometry and mathematical physics. Originating from string theory, the project will not only ....Integrable systems from gauge theories. This project aims to construct and describe three important integrable systems and to produce new significant results in representation theory and mathematical physics. The focus of the project is the theory of integrable systems, a breakthrough area in modern mathematics, and it will use tools from geometric representation theory to study the interrelation between geometry and mathematical physics. Originating from string theory, the project will not only provide a deeper understanding of the universe, it will also train young mathematicians and other highly qualified individuals with the potential to make a significant impact to technology, security, and the economy though their specialised skills.Read moreRead less
Logarithmic conformal field theory and the 4D/2D correspondence. Conformal field theory provides powerful methods for attacking problems in theoretical physics and furnishes beautiful connections between seemingly disparate branches of pure mathematics. This proposal aims to greatly expand our knowledge of the logarithmic conformal field theories that have recently witnessed a resurgence of interest in physics. Advancing these theories is crucial to progress in high-energy physics and pure mathe ....Logarithmic conformal field theory and the 4D/2D correspondence. Conformal field theory provides powerful methods for attacking problems in theoretical physics and furnishes beautiful connections between seemingly disparate branches of pure mathematics. This proposal aims to greatly expand our knowledge of the logarithmic conformal field theories that have recently witnessed a resurgence of interest in physics. Advancing these theories is crucial to progress in high-energy physics and pure mathematics. Expected outcomes include a completely new understanding of the mathematical structure of these theories which will, in turn, facilitate applications in 4D gauge theory. This will boost research capacity and further cement Australia's reputation as an international leader in mathematical physics research.Read moreRead less
New directions in Hecke algebras. To goal of this project is to make fundamental advances in representation theory, a powerful branch of mathematics focused on taking abstract mathematical structures and ``representing'' them in a concrete and useful way. In particular we aim to prove a series of long standing and influential conjectures by George Lusztig concerning the representation theory of Hecke algebras, objects which are ubiquitous in modern algebra. Our work will lead to new discoveries, ....New directions in Hecke algebras. To goal of this project is to make fundamental advances in representation theory, a powerful branch of mathematics focused on taking abstract mathematical structures and ``representing'' them in a concrete and useful way. In particular we aim to prove a series of long standing and influential conjectures by George Lusztig concerning the representation theory of Hecke algebras, objects which are ubiquitous in modern algebra. Our work will lead to new discoveries, a fundamentally deeper understanding of Kazhdan-Lusztig theory, and will drive future research. Benefits include enhanced international collaboration and increasing capacity in pure mathematics, especially in the cutting-edge area of representation theory.Read moreRead less
Australian Laureate Fellowships - Grant ID: FL170100032
Funder
Australian Research Council
Funding Amount
$2,837,520.00
Summary
Zero-dimensional symmetry and its ramifications. This project aims to investigate algebraic objects known as 0-dimensional groups, which are a mathematical tool for analysing the symmetry of infinite networks. Group theory has been used to classify possible types of symmetry in various contexts for nearly two centuries now, and 0-dimensional groups are the current frontier of knowledge. The expected outcome of the project is that the understanding of the abstract groups will be substantially adv ....Zero-dimensional symmetry and its ramifications. This project aims to investigate algebraic objects known as 0-dimensional groups, which are a mathematical tool for analysing the symmetry of infinite networks. Group theory has been used to classify possible types of symmetry in various contexts for nearly two centuries now, and 0-dimensional groups are the current frontier of knowledge. The expected outcome of the project is that the understanding of the abstract groups will be substantially advanced, and that this understanding will shed light on structures possessing 0-dimensional symmetry. In addition to being cultural achievements in their own right, advances in group theory such as this also often have significant translational benefits. This will provide benefits such as the creation of tools relevant to information science and researchers trained in the use of these tools.Read moreRead less
Algebraic Schubert geometry and unitary reflection groups. This project aims to generalise the recent work of Elias and Williamson to the complex case. Fundamental to the study of symmetry are the ubiquitous Coxeter groups, which have an associated set of critically important ‘Kazhdan-Lusztig polynomials’. For some Coxeter groups, these may be interpreted in terms of classical geometry, leading to deep positivity properties for their coefficients. Elias and Williamson have recently shown that th ....Algebraic Schubert geometry and unitary reflection groups. This project aims to generalise the recent work of Elias and Williamson to the complex case. Fundamental to the study of symmetry are the ubiquitous Coxeter groups, which have an associated set of critically important ‘Kazhdan-Lusztig polynomials’. For some Coxeter groups, these may be interpreted in terms of classical geometry, leading to deep positivity properties for their coefficients. Elias and Williamson have recently shown that this geometry may be simulated algebraically for any Coxeter group, so positivity for Kazhdan-Lusztig polynomials holds for all Coxeter groups. This result has explosive consequences in many areas of geometry and algebra. This project is designed to extend these results to complex unitary reflection groups, with potentially dramatic consequences in number theory, representation theory and topology.Read moreRead less
Graded semisimple deformations. Recent advances in representation theory have revealed beautiful new structures in the classical representation theory of the symmetric groups and Hecke algebras. These discoveries have provided us with new algebras, the cyclotomic KLR algebras, that encode deep properties of fundamental objects in algebraic combinatorics and geometric representation theory. The cyclotomic quiver Hecke algebras are central to several open problems in mathematics but they are still ....Graded semisimple deformations. Recent advances in representation theory have revealed beautiful new structures in the classical representation theory of the symmetric groups and Hecke algebras. These discoveries have provided us with new algebras, the cyclotomic KLR algebras, that encode deep properties of fundamental objects in algebraic combinatorics and geometric representation theory. The cyclotomic quiver Hecke algebras are central to several open problems in mathematics but they are still poorly understood, with even basic properties like their dimensions being unknown. This project will establish a new framework for studying these algebras that will remove the current obstacles in this field and alllow us to prove substantial new results that advance the theory.Read moreRead less
Computing with Lie groups and algebras: nilpotent orbits and applications. This project aims to advance knowledge of an important class of Lie algebras, for which recent work has shown that a deeper mathematical theory, and better computational tools are required. Lie theory is a mathematical area with rich applications in the physical sciences. Expected outcomes from this project include the first systematic treatment of these algebras, new powerful algorithms to compute with them, and explicit ....Computing with Lie groups and algebras: nilpotent orbits and applications. This project aims to advance knowledge of an important class of Lie algebras, for which recent work has shown that a deeper mathematical theory, and better computational tools are required. Lie theory is a mathematical area with rich applications in the physical sciences. Expected outcomes from this project include the first systematic treatment of these algebras, new powerful algorithms to compute with them, and explicit nilpotent orbit classifications that will solve open problems in black hole theory. This should significantly enhance fundamental mathematical research and the Lie functionality of leading computer algebra systems, and is expected to strengthen international linkages.Read moreRead less