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.
Discovery Early Career Researcher Award - Grant ID: DE200100407
Funder
Australian Research Council
Funding Amount
$427,066.00
Summary
Homology theories in quantum topology. This project aims to resolve a major 25-year-old open problem relating the quantum topology of knots, 3- and 4-dimensional spaces to higher representation theory, the study of hidden symmetries of algebraic structures. The project expects to use homological invariants of knots and the higher representation theory of quantum groups to construct highly anticipated invariants of 3- and 4-dimensional manifolds and tools to compute these invariants by reduction ....Homology theories in quantum topology. This project aims to resolve a major 25-year-old open problem relating the quantum topology of knots, 3- and 4-dimensional spaces to higher representation theory, the study of hidden symmetries of algebraic structures. The project expects to use homological invariants of knots and the higher representation theory of quantum groups to construct highly anticipated invariants of 3- and 4-dimensional manifolds and tools to compute these invariants by reduction to basic building blocks. Expected outcomes also include new connections to diverse areas in mathematics. This is expected to benefit Australian science by invigorating collaboration in mathematics and theoretical physics and by attracting students and distinguished research visitors. Read moreRead less
Enriched higher category theory. At the beginning of 21st century it became clear that further progress in mathematics and physics required a development of a new powerful language, which received the name Higher Category Theory. This project is devoted to the development of this new exciting theory.
From topological Hochschild homology to algebraic K-theory. The project will use methods of algebraic topology, specifically topological Hochschild homology, to study algebraic K-theory. This will increase our understanding of algebraic geometry, number theory, and the geometry of manifolds.
Flag varieties and configuration spaces in algebra. School students learn that curves may be described by means of equations, which may therefore be solved geometrically; this is an example of the interaction of algebra and geometry. In this project geometric ideas such as simplicial geometry and cohomological representation theory will be developed, which address deep questions in modern algebra.
Australian Laureate Fellowships - Grant ID: FL100100137
Funder
Australian Research Council
Funding Amount
$1,868,132.00
Summary
Derived categories and applications. This project will deepen our understanding of homological algebra, a mathematical tool that has proved useful in areas ranging from physics to the coding of information for computer transmission. Also, having a thriving research presence in Australia, of this vibrant, modern field, should inspire more students to seek a career in mathematics; this would help relieve the acute, well-documented shortage of mathematicians in Australia. It has been established th ....Derived categories and applications. This project will deepen our understanding of homological algebra, a mathematical tool that has proved useful in areas ranging from physics to the coding of information for computer transmission. Also, having a thriving research presence in Australia, of this vibrant, modern field, should inspire more students to seek a career in mathematics; this would help relieve the acute, well-documented shortage of mathematicians in Australia. It has been established that Australia is not producing enough mathematicians to meet the needs of industry; a lively centre, full of young, productive mathematicians, will go a long way towards correcting this problem.Read moreRead less
Generalised topological spaces. Pure mathematics creates abstractions of real-world entities; one such is the idea of a 'topological space', which abstracts from geometric forms like cubes and toruses. But topological spaces fail to capture geometric structures arising in areas like quantum physics; and this project seeks to rectify this, by developing a new more general notion.
Discovery Early Career Researcher Award - Grant ID: DE120100232
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Fusion categories and topological quantum field theory. This project will involve mathematical research of the highest international calibre on fusion categories and topological field theory. Progress in these fields will lead to advances in computing (for example substrates for quantum computers), condensed matter physics, and the mathematical fields of operator algebra, quantum algebra, and quantum topology.
Noncommutative analysis for self-similar structure. This project in pure mathematics aims to develop novel mathematical techniques for understanding self-similar structures using operator algebras. Fractals and self-similarity have many applications both within and outside mathematics, but remain deeply mysterious, while operator algebras are the mathematical language of quantum mechanics. This project expects to provide new connections between self similarity and operator algebras advancing bot ....Noncommutative analysis for self-similar structure. This project in pure mathematics aims to develop novel mathematical techniques for understanding self-similar structures using operator algebras. Fractals and self-similarity have many applications both within and outside mathematics, but remain deeply mysterious, while operator algebras are the mathematical language of quantum mechanics. This project expects to provide new connections between self similarity and operator algebras advancing both fields. Expected outcomes include increased understanding of self-similar structures, and novel operator-algebraic phenomena and examples. Benefits include growing Australia's capacity in operator algebras and mathematics more generally, and enhanced international collaboration.Read moreRead less
Quantum invariants and hyperbolic manifolds in three-dimensional topology. The project aims to broaden our understanding of three-dimensional (3-D) spaces, including spaces that arise in engineering, microbiology and physics. It is known that all 3-D spaces can be decomposed into geometric pieces. The most common type of geometry is hyperbolic. It is also known that such spaces have algebraic structures arising from quantum physics, known as quantum invariants. Several important conjectures, bas ....Quantum invariants and hyperbolic manifolds in three-dimensional topology. The project aims to broaden our understanding of three-dimensional (3-D) spaces, including spaces that arise in engineering, microbiology and physics. It is known that all 3-D spaces can be decomposed into geometric pieces. The most common type of geometry is hyperbolic. It is also known that such spaces have algebraic structures arising from quantum physics, known as quantum invariants. Several important conjectures, based on developments in physics, assert that hyperbolic geometry and quantum invariants are deeply related, but they remain unproved. The project aims to find new relationships between hyperbolic geometry and quantum invariants, advancing our understanding of both areas.Read moreRead less