Discovery Early Career Researcher Award - Grant ID: DE170100623
Funder
Australian Research Council
Funding Amount
$360,000.00
Summary
Quasi-hereditary categories in Lie theory. This project aims to use diagram algebras and categorical representation theory to study fundamental open problems in the representation theory of Lie algebras and their generalisations. The concept of symmetry is omnipresent in science and culture. Its mathematical study leads to the notion of groups, algebras and their representation theory. Representation theory is applicable in many active research areas, including subatomic particle physics and qua ....Quasi-hereditary categories in Lie theory. This project aims to use diagram algebras and categorical representation theory to study fundamental open problems in the representation theory of Lie algebras and their generalisations. The concept of symmetry is omnipresent in science and culture. Its mathematical study leads to the notion of groups, algebras and their representation theory. Representation theory is applicable in many active research areas, including subatomic particle physics and quantum computing. Solutions to these problems could lead to better understanding of several categories of representations of Lie algebras, and create new research tools.Read moreRead less
Symmetry via braiding, diagrammatics and cellularity. Symmetry is a basic organising tool for humans to understand their environment. Invariants are the mathematical embodiment of symmetry, and their study is as ancient as thought itself. This project aims to use the tools of braided tensor categories and cellular structure, to analyse the invariants occurring in several fundamental areas of mathematics, particularly relating to physics. The endomorphism algebras in certain tensor categories, pa ....Symmetry via braiding, diagrammatics and cellularity. Symmetry is a basic organising tool for humans to understand their environment. Invariants are the mathematical embodiment of symmetry, and their study is as ancient as thought itself. This project aims to use the tools of braided tensor categories and cellular structure, to analyse the invariants occurring in several fundamental areas of mathematics, particularly relating to physics. The endomorphism algebras in certain tensor categories, particularly those for quantised superalgebras, will be realised as diagram algebras, and analysed using cellular theory. The intended output include criteria for semisimplicity, a new theory of diagram algebras, and decomposition theory which are expected to permit the determination of multiplicities of composition factors.Read moreRead less
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.