Constructive representation theory of classical and quantum Lie superalgebras. Classical and quantum Lie superalgebras lie at the heart of many recent theoretical developments in the fields of integrable models and conformal field theory. Based on results published in 2013 by the Chief Investigators, it is evident that the time is right to further develop these ideas into a coherent and canonical framework. This ambitious and thorough proposal is focussed on solving sophisticated, contemporary p ....Constructive representation theory of classical and quantum Lie superalgebras. Classical and quantum Lie superalgebras lie at the heart of many recent theoretical developments in the fields of integrable models and conformal field theory. Based on results published in 2013 by the Chief Investigators, it is evident that the time is right to further develop these ideas into a coherent and canonical framework. This ambitious and thorough proposal is focussed on solving sophisticated, contemporary problems in representation theory related to classical and quantum Lie superalgebras that will have immediate consequences in these burgeoning fields.Read moreRead less
The Ricci curvature of homogeneous spaces. The geometry of homogeneous spaces is an area of research with applications in numerous fields, including topology, harmonic analysis, relativity and quantum theory. This project aims to resolve a fundamental problem in this area, known as the prescribed Ricci curvature problem for homogeneous metrics, and to settle the important and closely related question of Ricci iteration existence and convergence. Moreover, the project aims to exploit the interpla ....The Ricci curvature of homogeneous spaces. The geometry of homogeneous spaces is an area of research with applications in numerous fields, including topology, harmonic analysis, relativity and quantum theory. This project aims to resolve a fundamental problem in this area, known as the prescribed Ricci curvature problem for homogeneous metrics, and to settle the important and closely related question of Ricci iteration existence and convergence. Moreover, the project aims to exploit the interplay between geometry and algebra to provide new insight into the physically significant problem of classifying unitary Lie algebra representations. This project is expected to facilitate interdisciplinary interaction leading to exciting developments across a range of fields.Read moreRead less
Lie superalgebra representations: a geometric approach. The concept of a Lie group provides a mathematical underpinning for the idea of symmetry in mathematics, physics and chemistry. The project aims to advance two fundamental problems related to this concept: classification of unitary representations of Lie superalgebras, and the prescribed Ricci curvature problem on Lie groups. The research builds on newly-discovered connections between these problems to achieve exciting progress in their res ....Lie superalgebra representations: a geometric approach. The concept of a Lie group provides a mathematical underpinning for the idea of symmetry in mathematics, physics and chemistry. The project aims to advance two fundamental problems related to this concept: classification of unitary representations of Lie superalgebras, and the prescribed Ricci curvature problem on Lie groups. The research builds on newly-discovered connections between these problems to achieve exciting progress in their resolution. Outcomes are expected to find applications across a range of fields, such as condensed matter physics, particle physics, quantum field theory and knot theory. Anticipated benefits include stronger links between different areas of science achieved through a deeper understanding of symmetry.Read moreRead less
Macdonald polynomials: Combinatorics and representations. This proposal is part of the aim to build a world class research team in algebraic combinatorics and combinatorial representation theory at the University of Melbourne, led by the two CI.
These fields are currently experiencing very rapid growth and development, and a strong Australia based team will further enhance the country's strong reputation in combinatorics and algebra.
The project will also provide a perfect training ground fo ....Macdonald polynomials: Combinatorics and representations. This proposal is part of the aim to build a world class research team in algebraic combinatorics and combinatorial representation theory at the University of Melbourne, led by the two CI.
These fields are currently experiencing very rapid growth and development, and a strong Australia based team will further enhance the country's strong reputation in combinatorics and algebra.
The project will also provide a perfect training ground for Higher Degree Students with interests in pure mathematics as well as computer
algebra and symbolic computation.Read moreRead less
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
Geodetic groups: foundational problems in algebra and computer science. The project aims to resolve important and longstanding open problems in Geometric Group Theory and Theoretical Computer Science. Since the 1980s researchers have conjectured that the geometric property of being geodetic is equivalent to several purely algebraic, algorithmic, and language-theoretic characterisations.
The project team's expertise in geodesic properties of groups, the interaction between formal languages and g ....Geodetic groups: foundational problems in algebra and computer science. The project aims to resolve important and longstanding open problems in Geometric Group Theory and Theoretical Computer Science. Since the 1980s researchers have conjectured that the geometric property of being geodetic is equivalent to several purely algebraic, algorithmic, and language-theoretic characterisations.
The project team's expertise in geodesic properties of groups, the interaction between formal languages and groups, and the theory of rewriting systems, together with recent breakthroughs by the team ensures that significant results can be expected.
Benefits include training research students and postdoctoral researchers in cutting-edge techniques, and advancing fundamental knowledge in mathematics and computer science.Read moreRead less
Algorithms and computation in four-dimensional topology. This project will establish Australia as a world leader in computational topology, particularly in the all-important areas of topology in three and four dimensions. In four dimensions this work will be truly groundbreaking; until now the field has seen little development due to the complexity of the algorithms and computations required, and the applicant is in the unique position of having the necessary tools to make significant progress ....Algorithms and computation in four-dimensional topology. This project will establish Australia as a world leader in computational topology, particularly in the all-important areas of topology in three and four dimensions. In four dimensions this work will be truly groundbreaking; until now the field has seen little development due to the complexity of the algorithms and computations required, and the applicant is in the unique position of having the necessary tools to make significant progress in a feasible time frame. In three dimensions this project will strengthen the distinguished computational topology community in Melbourne, led by pioneers such as Rubinstein, Goodman, Hodgson as well as the applicant himself.Read moreRead less
Geometric partial differential systems and their applications. This proposal addresses questions central to the understanding of nonlinear partial differential systems from classical, quantum field theory and liquid crystals. Applications to physical problems such as the Yang-Mills flow, Faddeev's model and liquid crystal systems are of great interest and importance in the broader scientific community. The project will yield internationally significant results in theoretical mathematics, with ....Geometric partial differential systems and their applications. This proposal addresses questions central to the understanding of nonlinear partial differential systems from classical, quantum field theory and liquid crystals. Applications to physical problems such as the Yang-Mills flow, Faddeev's model and liquid crystal systems are of great interest and importance in the broader scientific community. The project will yield internationally significant results in theoretical mathematics, with applications in physics and and other sciences. Specialist training will be provided for Australia's next generation of mathematicians. This project will enable Australian researchers to stay at the forefront of research in this area, strengthening links with a number of world-leading mathematicians.Read moreRead less
Geometric variational problems and nonlinear partial differential systems. We will investigate several important problems on non-linear partial differential systems, bridging analysis, differential geometry and mathematical physics. Harmonic maps are the prototype of maps minimizing the Dirichlet energy. The liquid crystal configuration generalizes the harmonic map with values into two dimensional spheres. The Yang-Mills equations originated from particle physics. We will make fundamental contri ....Geometric variational problems and nonlinear partial differential systems. We will investigate several important problems on non-linear partial differential systems, bridging analysis, differential geometry and mathematical physics. Harmonic maps are the prototype of maps minimizing the Dirichlet energy. The liquid crystal configuration generalizes the harmonic map with values into two dimensional spheres. The Yang-Mills equations originated from particle physics. We will make fundamental contributions to these topics: Regularity problem and energy minimality of weakly harmonic maps, Weak solutions of the liquid crystal equilibrium system, Yang-Mills heat flow and singular Yang-Mills connections.
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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