Discovery Early Career Researcher Award - Grant ID: DE120102369
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Higher representation theory. Representation theory lies at the very centre of mathematics, with applications in all areas of mathematics and mathematical physics; at some level it is about observing the symmetries of a system and exploiting them, and this has been invaluable. This project will explore the forefront of the modern, higher version of this research field.
Flag varieties and configuration spaces in algebra. School students learn that curves may be described by means of equations, which may therefore be solved geometrically; this is an example of the interaction of algebra and geometry. In this project geometric ideas such as simplicial geometry and cohomological representation theory will be developed, which address deep questions in modern algebra.
Constructive Representation Theory. A group is a mathematical structure that captures the notion of symmetry. This project will enable us to perform a deep analysis of all the ways in which the group can act on an object such as a molecule, by constructing all of its representations as a matrix group.
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
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
Algebraic evolution and evolutionary algebra. Algebra and biology have developed in extraordinary ways over the last half century yet, to date, the use of algebraic ideas in biology has been limited. This project will address this by modelling evolutionary processes in bacteria using algebraic ideas.
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
The dimension problem for Hecke algebras. This project aims to give important new information about the graded Specht modules and the irreducible graded modules of the cyclotomic Hecke algebras. Experts have long considered that computing the dimensions of the irreducible representations to be completely intractable, however, the powerful new tools provided by the recently discovered KLR-grading gives rise to the combinatorics for solving this problem and for describing the graded decomposition ....The dimension problem for Hecke algebras. This project aims to give important new information about the graded Specht modules and the irreducible graded modules of the cyclotomic Hecke algebras. Experts have long considered that computing the dimensions of the irreducible representations to be completely intractable, however, the powerful new tools provided by the recently discovered KLR-grading gives rise to the combinatorics for solving this problem and for describing the graded decomposition numbers of these algebras. Even in characteristic zero this is incredibly interesting because, as a special case, it would give explicit combinatorial formulas for parabolic Kazhdan-Lusztig polynomials, a problem that has been studied intensely (without solution) for over thirty years.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE160100975
Funder
Australian Research Council
Funding Amount
$307,536.00
Summary
Algebraic groups and Springer theory. This project aims to explore representation theory, which is a study of the basic symmetries that occur in nature. By its nature, representation theory has applications to number theory, physics, national security and internet security, and other sciences. Generalised Springer theory plays an important role in representations of finite groups of Lie type. The project aims to develop an analogous theory in a more general setting that includes symmetric spaces ....Algebraic groups and Springer theory. This project aims to explore representation theory, which is a study of the basic symmetries that occur in nature. By its nature, representation theory has applications to number theory, physics, national security and internet security, and other sciences. Generalised Springer theory plays an important role in representations of finite groups of Lie type. The project aims to develop an analogous theory in a more general setting that includes symmetric spaces. Moreover, the project aims to address various outstanding problems in algebraic groups. The project also plans to explore the connection between the geometry of certain null-cones and deformations of Galois representations.Read moreRead less
Graded representations of Hecke algebras. This project makes fundamental contributions to the representation theory of Hecke algebras with applications to quantum groups, knot theory, Lie groups and topological quantum field theory. Building on exciting developments we will produce new gradings and then use them to tackle difficult questions in the representation theory of these algebras.