Discovery Early Career Researcher Award - Grant ID: DE120100173
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
A new upper bound for the Riemann zeta-function and applications to the distribution of prime numbers. Prime numbers are known to every schoolchild and are ubiquitous in modern cryptography; some of their deepest properties relate to a function called the Riemann zeta-function. This project aims at better estimating this function, thereby improving current knowledge on the distribution of prime numbers.
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
Representations of arithmetic groups and their associated zeta functions. This project aims to investigate deep connections between number theory and group theory by studying linear actions of arithmetic groups. Arithmetic groups are used in geometry, dynamics, number theory and other areas of pure mathematics. This project will study their representations from two perspectives. First, it will establish properties of the associated zeta functions to resolve open problems about the asymptotic beh ....Representations of arithmetic groups and their associated zeta functions. This project aims to investigate deep connections between number theory and group theory by studying linear actions of arithmetic groups. Arithmetic groups are used in geometry, dynamics, number theory and other areas of pure mathematics. This project will study their representations from two perspectives. First, it will establish properties of the associated zeta functions to resolve open problems about the asymptotic behaviour of the dimensions of the irreducible representations. Second, it will explore the evolution of representations across families of groups under new induction and restriction functors, in analogy with creation and annihilation operators in physics. The project will enhance Australia's capacity in representation theory and group theory, the mathematics that underline symmetry in nature.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
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
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
Groups, piecewise linear representations, and linear 2-representations. This project aims to address fundamental questions at the interface of two central areas of modern mathematics, geometric group theory and higher representation theory. Higher representation theory is a relatively new field, but its tools have already had a tremendous impact on mathematics. The project is expected to use these tools to address outstanding questions in geometric group theory. The expected outcomes of this pr ....Groups, piecewise linear representations, and linear 2-representations. This project aims to address fundamental questions at the interface of two central areas of modern mathematics, geometric group theory and higher representation theory. Higher representation theory is a relatively new field, but its tools have already had a tremendous impact on mathematics. The project is expected to use these tools to address outstanding questions in geometric group theory. The expected outcomes of this project include resolutions of open problems in the theory of Artin groups and the creation of a new subject, the dynamical theory of two-linear groups.Read moreRead less