Elliptic special functions. Although elliptic functions and special functions are both classical areas of mathematics, the field of elliptic special functions was only established in the last two decades. It combines ideas from analysis, modular forms and statistical mechanics to tackle problems in number theory (elliptic curves), algebra (elliptic quantum groups), mathematical physics (Seiberg duality) and more. This project aims to settle two important problems in the field of elliptic special ....Elliptic special functions. Although elliptic functions and special functions are both classical areas of mathematics, the field of elliptic special functions was only established in the last two decades. It combines ideas from analysis, modular forms and statistical mechanics to tackle problems in number theory (elliptic curves), algebra (elliptic quantum groups), mathematical physics (Seiberg duality) and more. This project aims to settle two important problems in the field of elliptic special functions: the resolution of Boyd's conjectures concerning Mahler measures and L-values of elliptic curves, and the construction of an Askey-Wilson-Koorwinder theory of elliptic biorthogonal functions for the A-type root system.Read moreRead less
ROBUST SOLID OXIDE FUEL CELL TECHNOLOGY FOR SMALL-SCALE APPLICATIONS. The project aims to develop nano-materials for the next generation planar Solid Oxide Fuel Cell (SOFC) that will operate at temperatures between 600 and 800°C. The goal is to identify and demonstrate materials that meet the robust requirements for small scale power generators at the 3-5kW scale. It is expected that these will be used in stationary power generation applications, in remote area power supplies, and for providing ....ROBUST SOLID OXIDE FUEL CELL TECHNOLOGY FOR SMALL-SCALE APPLICATIONS. The project aims to develop nano-materials for the next generation planar Solid Oxide Fuel Cell (SOFC) that will operate at temperatures between 600 and 800°C. The goal is to identify and demonstrate materials that meet the robust requirements for small scale power generators at the 3-5kW scale. It is expected that these will be used in stationary power generation applications, in remote area power supplies, and for providing auxiliary power in vehicles. The work builds on the world-leading position that Ceramic Fuel Cells Ltd. has in planar SOFC technology, utilising micro-analysis and fuel cell expertise at the University of Queensland.Read moreRead less
Towards logarithmic representation theory of W-algebras. Aims: To construct and analyse indecomposable representations of significance in conformal field theory.
Significance: Conformal field theory plays a key role in many developments in mathematics and physics. Logarithmic conformal field theories govern important systems such as two-dimensional critical percolation. This proposal aims to develop the representation theory necessary for understanding salient features of critical systems des ....Towards logarithmic representation theory of W-algebras. Aims: To construct and analyse indecomposable representations of significance in conformal field theory.
Significance: Conformal field theory plays a key role in many developments in mathematics and physics. Logarithmic conformal field theories govern important systems such as two-dimensional critical percolation. This proposal aims to develop the representation theory necessary for understanding salient features of critical systems described by logarithmic conformal field theory.
Expected Outcomes: Novel representations of fundamental importance in logarithmic conformal field theory.
Benefit: Resolution of open problems in logarithmic conformal field theory, thus continuing the strong tradition in the field in Australia.
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Categorical symmetries in representation theory. This project aims to develop categorical symmetries of central objects in mathematics such as braid groups, the Hilbert scheme of points, and the Virasoro algebra. The concept of symmetry is an important organising principle in science. Representation theory is the field of mathematics concerned with studying symmetries. The problems proposed have connections to many different areas including algebra, geometry, topology, and mathematical physics. ....Categorical symmetries in representation theory. This project aims to develop categorical symmetries of central objects in mathematics such as braid groups, the Hilbert scheme of points, and the Virasoro algebra. The concept of symmetry is an important organising principle in science. Representation theory is the field of mathematics concerned with studying symmetries. The problems proposed have connections to many different areas including algebra, geometry, topology, and mathematical physics. This project expects to advance pure mathematics and provide potential benefit in many related fields.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE200101802
Funder
Australian Research Council
Funding Amount
$354,016.00
Summary
Combinatorial and Representation Theoretic Methods in Number Theory. This Project aims to explore connections of Number Theory and Representation Theory by utilising tools of Algebraic Combinatorics. Symmetries and constructions of crucial number theoretic objects such as Whittaker functions are underpinned by models for Lie algebras and root systems. The Project expects to advance the algebraic framework of the constructions. Expected outcomes include a unified combinatorial model of these obje ....Combinatorial and Representation Theoretic Methods in Number Theory. This Project aims to explore connections of Number Theory and Representation Theory by utilising tools of Algebraic Combinatorics. Symmetries and constructions of crucial number theoretic objects such as Whittaker functions are underpinned by models for Lie algebras and root systems. The Project expects to advance the algebraic framework of the constructions. Expected outcomes include a unified combinatorial model of these objects, and an extension of the costructions to the infinite dimensional setting. This will benefit the applications in Number Theory and strengthen nascent connections with Mathematical Physics. Read moreRead less
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
Discovery Early Career Researcher Award - Grant ID: DE140100633
Funder
Australian Research Council
Funding Amount
$395,169.00
Summary
Problems in the Langlands Program. The Langlands program is an international research program sitting at the interface of number theory, representation theory, algebraic geometry, and mathematical physics. The aim of this project is to prove three conjectures in this program. Settling these conjectures would lead to significant advances in the Langlands program by strengthening connections between this program and the geometry of loop groups, representations of finite groups, and representations ....Problems in the Langlands Program. The Langlands program is an international research program sitting at the interface of number theory, representation theory, algebraic geometry, and mathematical physics. The aim of this project is to prove three conjectures in this program. Settling these conjectures would lead to significant advances in the Langlands program by strengthening connections between this program and the geometry of loop groups, representations of finite groups, and representations of affine Kac-Moody algebras at the critical level.Read moreRead less