Verifying the Riemann hypothesis to large heights: theory and applications. This project aims to verify the Riemann hypothesis to a record height and apply this verification to the distribution of prime numbers. The Riemann hypothesis (an open problem for 150 years) is ubiquitous in analytic number theory and prevalent in many other areas of mathematics. This project plans to use state-of-the-art computational hardware and the mathematical and algorithmic expertise of the investigators to verify ....Verifying the Riemann hypothesis to large heights: theory and applications. This project aims to verify the Riemann hypothesis to a record height and apply this verification to the distribution of prime numbers. The Riemann hypothesis (an open problem for 150 years) is ubiquitous in analytic number theory and prevalent in many other areas of mathematics. This project plans to use state-of-the-art computational hardware and the mathematical and algorithmic expertise of the investigators to verify the Riemann hypothesis several orders of magnitude further than what is currently known. A secondary aim is to apply this new verification to a multitude of results in analytic number theory: this would provide future researchers with vastly superior results.Read moreRead less
Braid groups and higher representation theory. Symmetry is a central notion in classical representation theory. In higher representation theory the symmetries of classical representation theory are replaced by higher symmetries. These higher symmetries contain new structure not present at the classical level. The proposed research will develop the higher representation theory of fundamental objects from classical representation theory and geometric group theory, focusing on braid groups and quan ....Braid groups and higher representation theory. Symmetry is a central notion in classical representation theory. In higher representation theory the symmetries of classical representation theory are replaced by higher symmetries. These higher symmetries contain new structure not present at the classical level. The proposed research will develop the higher representation theory of fundamental objects from classical representation theory and geometric group theory, focusing on braid groups and quantum groups.Read moreRead less
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
Algebraic K-theory and groups. This project will study the K-theory of division algebras, their generalisation to Azumaya algebras and the nonstable K-theory of rings. Expected outcomes would enhance our understanding on the structure of these K groups.
The goal is to settle some of the most significant conjectures in the subject: Bak's solvability of nonstable K groups over rings and the Merkurjev-Suslin conjectures on reduced K theory of division rings.
The study of these problems contribu ....Algebraic K-theory and groups. This project will study the K-theory of division algebras, their generalisation to Azumaya algebras and the nonstable K-theory of rings. Expected outcomes would enhance our understanding on the structure of these K groups.
The goal is to settle some of the most significant conjectures in the subject: Bak's solvability of nonstable K groups over rings and the Merkurjev-Suslin conjectures on reduced K theory of division rings.
The study of these problems contributes to and draws from such topics as group theory, commutative ring theory, algebraic number theory and algebraic geometry.
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Low dimensional categories. This project plans to study fundamental examples of higher categories in dimensions 2, 3, and 4, with the goal of understanding their essential features and building appropriate tools and theoretical frameworks for working with them. This work would have applications in several areas of mathematics including representation theory, low dimensional topology and topological quantum computing. Higher categories let us study the possible shapes of space (dimensions 2, 3, a ....Low dimensional categories. This project plans to study fundamental examples of higher categories in dimensions 2, 3, and 4, with the goal of understanding their essential features and building appropriate tools and theoretical frameworks for working with them. This work would have applications in several areas of mathematics including representation theory, low dimensional topology and topological quantum computing. Higher categories let us study the possible shapes of space (dimensions 2, 3, and 4, are the relevant ones for the world we live in), and also the dimensions in which we find the most interesting examples. The project plans to investigate particular examples related to exceptional Lie algebras, fusion categories, and categorical link invariants.Read moreRead less
Topics in triangulated categories. This project in pure mathematics, more specifically in modern homological algebra, builds on work started by the chief investigator in the last five years. What has already been done has achieved striking results, solving very different problems that have been open for two decades. And there seem to be many directions in which it could be pursued further.
The international mathematical community seems intrigued by what the chief investigator has achieved recen ....Topics in triangulated categories. This project in pure mathematics, more specifically in modern homological algebra, builds on work started by the chief investigator in the last five years. What has already been done has achieved striking results, solving very different problems that have been open for two decades. And there seem to be many directions in which it could be pursued further.
The international mathematical community seems intrigued by what the chief investigator has achieved recently - judging by invitations to give prestigious talks and the feedback at these events. The expected outcome is major progress in our understanding of derived categories, as well as diverse applications. The benefit will be to enhance the international stature of Australian science.Read moreRead less
Moduli, invariants, and algebraisation. This project is in pure mathematics. It aims to address gaps in our
knowledge in the modern geometries and their associated algebraic structures that arise in classification problems that pervade mathematics and its applications.
This project expects to generate new knowledge in modern algebra and geometry.
Expected outcomes of this project include major progress in our
understanding of invariants of derived categories of algebraic stacks and the
relat ....Moduli, invariants, and algebraisation. This project is in pure mathematics. It aims to address gaps in our
knowledge in the modern geometries and their associated algebraic structures that arise in classification problems that pervade mathematics and its applications.
This project expects to generate new knowledge in modern algebra and geometry.
Expected outcomes of this project include major progress in our
understanding of invariants of derived categories of algebraic stacks and the
relationship between algebraic and other geometries.
The benefit will be to enhance the international stature of Australian science.Read moreRead less
Algebraic and computational approaches for classical and quantum systems. This project aims to use a combination of algebraic, analytic and numerical techniques to develop computational algorithms to address a range of notoriously challenging problems in the mathematical sciences. These problems involve predicting the large-scale behaviour of strongly interacting classical and quantum spin systems originating in condensed matter physics, including models of relevance to proposals for topological ....Algebraic and computational approaches for classical and quantum systems. This project aims to use a combination of algebraic, analytic and numerical techniques to develop computational algorithms to address a range of notoriously challenging problems in the mathematical sciences. These problems involve predicting the large-scale behaviour of strongly interacting classical and quantum spin systems originating in condensed matter physics, including models of relevance to proposals for topological quantum computation and the latest progress using field theory. The project outcomes will involve advances in understanding these systems from new exact results and high precision numerical estimates.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