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: DE140101519
Funder
Australian Research Council
Funding Amount
$393,979.00
Summary
Advances in algebraic stacks and applications. Algebraic stacks are a geometric manifestation of algebraic and physical phenomena. Stacks provide a fundamental mathematical structure to study questions in geometry, topology and number theory having deep applications to string theory and complexity theory. This project will prove new fundamental theorems about algebraic stacks that will have broad implications. In particular, the new results obtained on algebraic stacks will be applied in order t ....Advances in algebraic stacks and applications. Algebraic stacks are a geometric manifestation of algebraic and physical phenomena. Stacks provide a fundamental mathematical structure to study questions in geometry, topology and number theory having deep applications to string theory and complexity theory. This project will prove new fundamental theorems about algebraic stacks that will have broad implications. In particular, the new results obtained on algebraic stacks will be applied in order to resolve a long-standing open problem in algebraic geometry. Specifically, the project will provide a new description of the birational geometry of one of the most interesting and studied algebraic varieties, the moduli space of smooth curves.Read moreRead less
Interactions between non-commutative algebra and algebraic geometry. Non-commutative algebra and algebraic geometry are both classical branches of maths. Recently, there has been an explosion of research involving intriguing interactions between the two rich disciplines, driven as much by considerations of physics (for example, string theory) as by maths itself. This project forms an integral part of these developments.
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
Discovery Early Career Researcher Award - Grant ID: DE150101799
Funder
Australian Research Council
Funding Amount
$315,000.00
Summary
Algebraic stacks through the Tannakian perspective. Algebraic stacks are natural types of spaces to consider when parameterising geometric objects in mathematics and physics. The Tannakian formalism allows one to view algebraic stacks through the way it acts on other geometric objects. This project aims to employ the perspective provided by the Tannakian formalism to prove innovative and foundational results in order to elucidate the geometry of algebraic stacks.
Towards a new concrete theory of cohomology: a fundamental concept in geometry. This project will develop a geometric linearisation method related to Witt vectors, an exotic but important number system. This will let us take one more step towards solving a fifty-year-old mystery: to find the elusive universal linearisation in algebraic geometry, which is the linearisation that controls all the others.
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
Big de Rham-Witt cohomology: towards a concrete theory of motives. This project will develop a geometric linearisation method related to Witt vectors, an exotic but important number system. This will let us take one more step towards solving a fifty-year-old mystery: to find the elusive universal linearisation in algebraic geometry, which is the linearisation that controls all the others.
Derived categories and their many applications. This project aims to work on major open problems in algebra, towards the solution of conjectures that have been around for decades. The proposed research is ground-breaking, introducing new methods to problems that have stumped experts around the world. The planned techniques the project will use come from homological algebra, more specifically, from derived categories. Preliminary work, using the new methods, has already led to major advances on w ....Derived categories and their many applications. This project aims to work on major open problems in algebra, towards the solution of conjectures that have been around for decades. The proposed research is ground-breaking, introducing new methods to problems that have stumped experts around the world. The planned techniques the project will use come from homological algebra, more specifically, from derived categories. Preliminary work, using the new methods, has already led to major advances on what was previously known. The project is ambitious: if really successful, it is hoped to reshape the subject and deepen our understanding of the field, but even more modest achievements are expected to clarify and improve on work by international experts over four decades.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