Algebraic categories and categorical algebra. Algebra is the study of operations, such as addition and multiplication, and the relationships between these operations. This project will study two exciting new branches of algebra, quantum algebra and postmodern algebra, which will lead to important advances in physics, geometry, and computing.
New constructions and techniques for tensor categories. The goal of this project is to make fundamental advances in the structure theory of tensor categories. Such categories play crucial roles in numerous fields of mathematics, physics and beyond. New methods, theory and examples will be developed, inspired by algebra, representation theory and geometry. These will then be applied in the foundational study of tensor categories for (dis)proving several of the most important open conjectures in t ....New constructions and techniques for tensor categories. The goal of this project is to make fundamental advances in the structure theory of tensor categories. Such categories play crucial roles in numerous fields of mathematics, physics and beyond. New methods, theory and examples will be developed, inspired by algebra, representation theory and geometry. These will then be applied in the foundational study of tensor categories for (dis)proving several of the most important open conjectures in the field. This will open new perspectives for applications in other areas, most notably in representation theory. Other benefits include enhanced international collaboration and scientific capacity in Australia.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE150101161
Funder
Australian Research Council
Funding Amount
$330,000.00
Summary
Quantum Groups and Categorification in Geometric Representation Theory. Representation theory, the mathematical study of symmetry, has applications in diverse areas such as particle physics, computer science, and molecular biology. This project aims to use a new family of quantum groups to prove a network of conjectures about categorical and geometric representation theory. The project aims to answer a long-standing open question in geometric representation theory concerning two families of geom ....Quantum Groups and Categorification in Geometric Representation Theory. Representation theory, the mathematical study of symmetry, has applications in diverse areas such as particle physics, computer science, and molecular biology. This project aims to use a new family of quantum groups to prove a network of conjectures about categorical and geometric representation theory. The project aims to answer a long-standing open question in geometric representation theory concerning two families of geometric spaces underlying the theory of Lie groups. Representation theory is a strength of Australian mathematics, and this project aims to undertake pressing research at the forefront of this dynamic field.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE190101099
Funder
Australian Research Council
Funding Amount
$420,256.00
Summary
Representation theory: studies of symmetry shadows. This project aims to solve fundamental problems in representation theory by combining cutting-edge techniques and developing novel higher level structures. Representation theory is the mathematical study of symmetry, an essential concept in science. Since the 1990s, mathematicians have been observing shadows of a more general notion of symmetry but so far have failed to explain it. Expected outcomes include a structural explanation of these sh ....Representation theory: studies of symmetry shadows. This project aims to solve fundamental problems in representation theory by combining cutting-edge techniques and developing novel higher level structures. Representation theory is the mathematical study of symmetry, an essential concept in science. Since the 1990s, mathematicians have been observing shadows of a more general notion of symmetry but so far have failed to explain it. Expected outcomes include a structural explanation of these shadows, new mathematical software to understand them and solutions to important conjectures. This project will make a significant contribution to the field of representation theory, with ramifications in mathematical physics and computer science.Read moreRead less
New dualities in modular representation theory. This project aims to introduce new geometric techniques in the modular representation theory of the general linear group, and resolve fundamental questions about modular representations of orthogonal and symplectic groups. Representation theory, the mathematical study of symmetry, has applications in many areas ranging from physics to computer science. The modular setting, where the characteristic of the field is positive, is a source of the deepes ....New dualities in modular representation theory. This project aims to introduce new geometric techniques in the modular representation theory of the general linear group, and resolve fundamental questions about modular representations of orthogonal and symplectic groups. Representation theory, the mathematical study of symmetry, has applications in many areas ranging from physics to computer science. The modular setting, where the characteristic of the field is positive, is a source of the deepest conjectures in this subject. Recent breakthroughs have provided approaches to some problems in modular representation theory that were previously inaccessible. This project will promote research at the forefront of the field, and help maintain Australia's international standing. It will train Australian postgraduates, who will be able to apply their research skills in industry, and strengthen the country's ties to a vibrant international community.Read moreRead less
New frontiers in the theory of noncommutative surfaces. In the 90s, Artin launched his school of noncommutative algebraic geometry, where novel geometric methods
were used to profoundly deepen our understanding of the classical subject of noncommutative algebra. This
project aims to advance this theory by establishing several new frontiers in the theory of noncommutative
surfaces. This project expects to develop new methods involving sheaf theory, Mori's minimal model program and
moduli stacks, ....New frontiers in the theory of noncommutative surfaces. In the 90s, Artin launched his school of noncommutative algebraic geometry, where novel geometric methods
were used to profoundly deepen our understanding of the classical subject of noncommutative algebra. This
project aims to advance this theory by establishing several new frontiers in the theory of noncommutative
surfaces. This project expects to develop new methods involving sheaf theory, Mori's minimal model program and
moduli stacks, to study in particular, Artin's classification problem for noncommutative surfaces. Expected
outcomes include a much richer geometric understanding of noncommutative algebra. This project should help
ensure Australia plays a leading role in important developments in both algebra and algebraic geometry.Read moreRead less
Groupoids as bridges between algebra and analysis. This pure mathematics project focuses on the interplay between abstract algebra and the area of functional analysis known as operator algebras. Specifically, it is intended to deal with generalisations of graph C*-algebras and of Leavitt path algebras. Over the last decade, researchers have discovered striking similarities between these areas, but no unifying result that would allow them to transfer techniques and theorems systematically from on ....Groupoids as bridges between algebra and analysis. This pure mathematics project focuses on the interplay between abstract algebra and the area of functional analysis known as operator algebras. Specifically, it is intended to deal with generalisations of graph C*-algebras and of Leavitt path algebras. Over the last decade, researchers have discovered striking similarities between these areas, but no unifying result that would allow them to transfer techniques and theorems systematically from one to the other. Recent research suggests that groupoid models for both algebras and C*-algebras may provide the missing link. This project aims to determine the role of groupoids in the two theories, and analyse and exploit the resulting synergies between abstract algebra and operator algebras.Read moreRead less
Graded K-theory as invariants for path algebras. This pure mathematics project focuses on Leavitt path algebras, which are structures that naturally arise from movements on directed graphs. These algebras appear in diverse areas (eg analysis, noncommutative geometry, representation theory and group theory). The aim of this project is to understand the behaviour of Leavitt path algebras and to classify them completely by means of graded K-theory. The project is an algebraic counterpart to graph C ....Graded K-theory as invariants for path algebras. This pure mathematics project focuses on Leavitt path algebras, which are structures that naturally arise from movements on directed graphs. These algebras appear in diverse areas (eg analysis, noncommutative geometry, representation theory and group theory). The aim of this project is to understand the behaviour of Leavitt path algebras and to classify them completely by means of graded K-theory. The project is an algebraic counterpart to graph C*-algebras (analytic structures that originated in Australian universities); both subjects have become areas of intensive research globally. The expected outcomes are to classify Leavitt path algebras, and to find a bridge (via graded K-theory) to graph C*-algebras and symbolic dynamics.Read moreRead less
Counting points on algebraic surfaces. This project aims to develop algorithms for calculating the number of solutions to polynomial equations and to compute zeta functions of certain types of algebraic varieties. Existing algorithms cannot solve these problems. The new algorithms will enable researchers in number theory to test and refine conjectures on generalisations of many famous problems, such as the Sato-Tate conjecture, the Lang-Trotter conjecture and the Birch-Swinnerton-Dyer conjecture ....Counting points on algebraic surfaces. This project aims to develop algorithms for calculating the number of solutions to polynomial equations and to compute zeta functions of certain types of algebraic varieties. Existing algorithms cannot solve these problems. The new algorithms will enable researchers in number theory to test and refine conjectures on generalisations of many famous problems, such as the Sato-Tate conjecture, the Lang-Trotter conjecture and the Birch-Swinnerton-Dyer conjecture. The project will also have a flow-on effect in other areas of mathematics and computer science where zeta functions play a central role, including cryptography, coding theory and mathematical physics.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE220100859
Funder
Australian Research Council
Funding Amount
$354,000.00
Summary
New techniques for exponential sums over low degree polynomials. This project aims to obtain new quantitative estimates for Weyl sums over low degree polynomials. Such estimates are fundamental to several areas of number theory. By interfacing techniques from diverse areas of mathematics, including algebraic geometry, analytic number theory, the geometry of numbers and harmonic analysis, this project will provide the first progress on estimating Weyl sums over low degree polynomials in over a ce ....New techniques for exponential sums over low degree polynomials. This project aims to obtain new quantitative estimates for Weyl sums over low degree polynomials. Such estimates are fundamental to several areas of number theory. By interfacing techniques from diverse areas of mathematics, including algebraic geometry, analytic number theory, the geometry of numbers and harmonic analysis, this project will provide the first progress on estimating Weyl sums over low degree polynomials in over a century. The expected outcomes include a deeper understanding of Weyl sums and enhanced international collaborations. Such progress will place Australia at the forefront of this important branch of number theory.Read moreRead less