Discovery Early Career Researcher Award - Grant ID: DE190101222
Funder
Australian Research Council
Funding Amount
$348,070.00
Summary
Elliptic representation theory: the study of symmetries across geometry, algebra and physics. This project aims to study symmetries which control geometry and physics of four-dimensional manifolds. Representation theory is the area in pure mathematics which studies symmetries coming from geometry, algebra, and physics. The expected outcome is new families of quantum algebra, their character formulas, and a solution to an infinite-dimensional moduli problem. This project will further deepen the m ....Elliptic representation theory: the study of symmetries across geometry, algebra and physics. This project aims to study symmetries which control geometry and physics of four-dimensional manifolds. Representation theory is the area in pure mathematics which studies symmetries coming from geometry, algebra, and physics. The expected outcome is new families of quantum algebra, their character formulas, and a solution to an infinite-dimensional moduli problem. This project will further deepen the mathematical knowledge of symmetries, and show unexpected new connections between different areas of pure mathematics and mathematical physics.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
Australian Laureate Fellowships - Grant ID: FL200100141
Funder
Australian Research Council
Funding Amount
$3,077,547.00
Summary
Real groups and the Langlands program. This program aims to address deep longstanding questions about real groups, algebraic objects which describe the basic symmetries occurring in nature. The study of these basic symmetries is central in all areas of mathematics and they come up in many applications. The expected outcomes include solving a central 50 year old problem of unitarity as well as making major progress in the Langlands program, a grand unification scheme of mathematics. The benefits ....Real groups and the Langlands program. This program aims to address deep longstanding questions about real groups, algebraic objects which describe the basic symmetries occurring in nature. The study of these basic symmetries is central in all areas of mathematics and they come up in many applications. The expected outcomes include solving a central 50 year old problem of unitarity as well as making major progress in the Langlands program, a grand unification scheme of mathematics. The benefits include raising Australia's international research profile, building a large network of international collaboration with top institutions in the world, and increasing capacity in number theory and algebraic geometry, which are playing an ever more important role in technology. Read moreRead less
Geometric methods in representation theory and the Langlands program. This research project aims to study questions in representation theory of groups using geometric methods. A central role is played by Langlands program which, broadly understood, can be viewed as a grand unified theory of mathematics. One setting for the work is modular representation theory with the aim of understanding irreducible characters. The project also aims to work on combinatorics and geometry in algebraic groups in ....Geometric methods in representation theory and the Langlands program. This research project aims to study questions in representation theory of groups using geometric methods. A central role is played by Langlands program which, broadly understood, can be viewed as a grand unified theory of mathematics. One setting for the work is modular representation theory with the aim of understanding irreducible characters. The project also aims to work on combinatorics and geometry in algebraic groups in small characteristics and one goal is to obtain a more uniform geometric understanding across all characteristics. The project also aims to work in the context of real groups and with the Gukov-Witten "fix of the orbit method" via branes. Finally, the project expects to begin a study of deformations of Galois representations in a general context.Read moreRead less
Algebraic invariants of singularities. This project aims to study the local and global behaviour of singularities that algebraic equations can describe via difficult algebraic invariants constructed from (algebraic) functions on the geometric object. A geometric object has a singularity at a point where its tangent directions do not behave the way they should. Examples include black holes, the vertex of a cone or a road intersection. This project is expected to contribute to fundamental research ....Algebraic invariants of singularities. This project aims to study the local and global behaviour of singularities that algebraic equations can describe via difficult algebraic invariants constructed from (algebraic) functions on the geometric object. A geometric object has a singularity at a point where its tangent directions do not behave the way they should. Examples include black holes, the vertex of a cone or a road intersection. This project is expected to contribute to fundamental research goals in pure mathematics, and increase the international competitiveness of Australian mathematics research.Read moreRead less
Topology through applications: geometry, number theory and physics. Topology is the part of geometry that remains invariant under deformation (as in the inflation of a balloon). We will apply this flexibility to investigate deep problems in several disciplines as diverse as number theory, geometry and the mathematics of string theory.