Discovery Early Career Researcher Award - Grant ID: DE190100888
Funder
Australian Research Council
Funding Amount
$333,924.00
Summary
Linear recurrence sequences over function fields and their applications. This project aims to deeply and systematically develop the theory of linear recurrence sequences (LRS) defined over function fields. Linear recurrence sequences (LRS) appear almost everywhere in mathematics and computer science. The project is expected to expand our knowledge on LRS and will span a wide range of new research directions. Through investigating and revealing the theoretical and practical aspects of LRS over fu ....Linear recurrence sequences over function fields and their applications. This project aims to deeply and systematically develop the theory of linear recurrence sequences (LRS) defined over function fields. Linear recurrence sequences (LRS) appear almost everywhere in mathematics and computer science. The project is expected to expand our knowledge on LRS and will span a wide range of new research directions. Through investigating and revealing the theoretical and practical aspects of LRS over function fields, the project will enrich the toolkits for cybersecurity by providing new approaches to cryptography. The outcomes of the project will help position Australia as a leader in this field.Read moreRead less
Elliptic curves: number theoretic and cryptographic aspects. Smart information use is of fundamental nature and has a great number of applications. First-generation security solutions are unable to support the modern requirements and new security infrastructures are emerging that must be carefully, but rapidly, defined. This urgently needs new mathematical tools, which is the main goal of this project.
The Mukhin-Varchenko and Rogers-Ramanujan conjectures. This project is aimed at proving two deep conjectures in pure mathematics. The conjectures are linked to many areas of mathematics, and success in proving either conjecture will signify a fundamental breakthrough in the fields of algebra, combinatorics and number theory.
Symmetric functions and Hodge polynomials. This project aims to explain a connection between two seemingly disparate mathematical notions: mixed Hodge polynomials of certain varieties, naturally arising in algebraic geometry, and Macdonald polynomials from the theory of symmetric functions. This project will resolve this connection using symmetric function theory, algebraic combinatorics and representation theory. This project could enhance Australia's international reputation in algebraic combi ....Symmetric functions and Hodge polynomials. This project aims to explain a connection between two seemingly disparate mathematical notions: mixed Hodge polynomials of certain varieties, naturally arising in algebraic geometry, and Macdonald polynomials from the theory of symmetric functions. This project will resolve this connection using symmetric function theory, algebraic combinatorics and representation theory. This project could enhance Australia's international reputation in algebraic combinatorics, combinatorial representation theory and algebraic geometry.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
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
Affine flags, folded galleries and euclidean reflection groups. This project aims to answer important geometric questions about subvarieties of the affine flag variety which are fundamental to algebraic geometry and number theory. It will answer basic questions about these central objects of mathematics, affine flags and their subspaces, using powerful new methods which combine ideas from geometry and algebra. The project expects to include finding the patterns of non-emptiness of these subvar ....Affine flags, folded galleries and euclidean reflection groups. This project aims to answer important geometric questions about subvarieties of the affine flag variety which are fundamental to algebraic geometry and number theory. It will answer basic questions about these central objects of mathematics, affine flags and their subspaces, using powerful new methods which combine ideas from geometry and algebra. The project expects to include finding the patterns of non-emptiness of these subvarieties and formulae for their dimension. It will develop and apply new methods which combine folded galleries and the geometry of Euclidean reflection groups, and these methods will have applications in algebraic combinatorics and representation theory. The project will also inspire productive connections between geometric group theory, a new and fast-growing area, and the classical fields of algebraic geometry, algebraic combinatorics and representation theory.Read moreRead less
The geometry and combinatorics of loop groups. This project deals with an exciting area at the very forefront of modern mathematics: the theory of MV-cycles. The project will provide new insights into the MV-cycles, and will strengthen Australia's position in this booming area.
Explicit methods in number theory: Computation, theory and application. This project aims to use explicit estimates to unify three problems in number theory: primitive roots, Diophantine quintuples, and linear independence of zeroes of the Riemann zeta-function. It will use computational and analytic number theory to reduce the quintuples problem to a soluble level. Pursuing relations between the zeta zeroes will overhaul many current results. This project will apply its findings about primitive ....Explicit methods in number theory: Computation, theory and application. This project aims to use explicit estimates to unify three problems in number theory: primitive roots, Diophantine quintuples, and linear independence of zeroes of the Riemann zeta-function. It will use computational and analytic number theory to reduce the quintuples problem to a soluble level. Pursuing relations between the zeta zeroes will overhaul many current results. This project will apply its findings about primitive roots to signal processing, cryptography and cybersecurity.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE150101415
Funder
Australian Research Council
Funding Amount
$360,000.00
Summary
Higher Representation Theory. Representation theory is a field of mathematics with applications across the breadth of mathematical study in fields as diverse as number theory and physics. The study of higher (or categorical) representation theory is a modern set of tools that provides new insights into representation theoretic phenomena. This project aims to study categorified quantum groups and, in particular, the categorifications provided by diagrammatic algebras. The project aims to further ....Higher Representation Theory. Representation theory is a field of mathematics with applications across the breadth of mathematical study in fields as diverse as number theory and physics. The study of higher (or categorical) representation theory is a modern set of tools that provides new insights into representation theoretic phenomena. This project aims to study categorified quantum groups and, in particular, the categorifications provided by diagrammatic algebras. The project aims to further develop the theory of Khovanov-Lauda-Rouquier (KLR) algebras, providing important foundational results for future research to build upon.Read moreRead less