Structured barrier and penalty functions in infinite dimensional optimisation and analysis. Very large scale tightly-constrained optimisation problems are ubiquitous and include water management, traffic flow, and imaging at telescopes and hospitals. Massively parallel computers can solve such problems and provide physically realisable solution only if subtle design issues are mastered. Resolving such issues is the goal of this project.
Discovery Early Career Researcher Award - Grant ID: DE160100525
Funder
Australian Research Council
Funding Amount
$392,053.00
Summary
Index Theory for Spaces with Symmetries. This project aims to study spaces with symmetries, which are important geometric models in topology and representation theory. The project plans to conduct research into geometric approaches to the representation theory of groups using KK theory and index theory from the perspective of operator algebra. The expected outcomes of this project are constructions of new topological invariants and their geometric formulas for spaces with symmetries and applicat ....Index Theory for Spaces with Symmetries. This project aims to study spaces with symmetries, which are important geometric models in topology and representation theory. The project plans to conduct research into geometric approaches to the representation theory of groups using KK theory and index theory from the perspective of operator algebra. The expected outcomes of this project are constructions of new topological invariants and their geometric formulas for spaces with symmetries and applications in representation theory.Read moreRead less
Supersymmetric quantum field theory, topology and duality. Supersymmetry is universally considered as one of the most fundamental concepts in physics, playing an increasingly central role in recent studies of quantum field theory and string theory. There is a corresponding development of supersymmetry in mathematics and this project will make advances both in 'superphysics' and 'supermathematics'.
Discovery Early Career Researcher Award - Grant ID: DE120100901
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Deformation quantisation and index theory for semi-simple groups. Deformation quantisation is a mathematical technique for describing the counter-intuitive geometry of quantum physics as a small variation of classical geometry as Newton would have known it. This project will apply the same techniques to solve fundamental mathematical problems in the study of symmetries.
Advances in index theory. The laws of nature are often expressed in terms of differential equations, which if elliptic, have an index being the number of solutions minus the number of constraints imposed. The Atiyah-Singer Index Theorem gives a striking calculation of this index and the projects involve innovative extensions of this theory with novel applications.
Ubiquity of K-theory and T-duality. An abstract mathematical tool, called K-theory, has recently found application in two, not obviously related, areas of physics: the classification of D-branes in String Theory, and topological phases in Condensed Matter Theory. This project aims to advance the development of K-theory using ideas from physics. In particular, the project aims to generalise previous constructions, such as T-duality, to loop spaces, and to develop the K-theory relevant to the clas ....Ubiquity of K-theory and T-duality. An abstract mathematical tool, called K-theory, has recently found application in two, not obviously related, areas of physics: the classification of D-branes in String Theory, and topological phases in Condensed Matter Theory. This project aims to advance the development of K-theory using ideas from physics. In particular, the project aims to generalise previous constructions, such as T-duality, to loop spaces, and to develop the K-theory relevant to the classification of topological phases in strongly interacting systems. This project involves postgraduate training as a crucial tool in achieving its aims and enhances Australia's position at the forefront of international research.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE170100149
Funder
Australian Research Council
Funding Amount
$357,000.00
Summary
T-duality and K-theory: Unity of condensed matter and string theory. This project aims to uncover deep mathematical structures which underlie recent discoveries at the forefront of string theory and condensed matter physics, using K-theory and T-duality as guiding themes. Inspired by string theory, T-duality techniques and geometric Fourier-Mukai transforms will be developed to study topological phases of matter. Similarly, topological materials motivate the detailed study of real twisted K-theo ....T-duality and K-theory: Unity of condensed matter and string theory. This project aims to uncover deep mathematical structures which underlie recent discoveries at the forefront of string theory and condensed matter physics, using K-theory and T-duality as guiding themes. Inspired by string theory, T-duality techniques and geometric Fourier-Mukai transforms will be developed to study topological phases of matter. Similarly, topological materials motivate the detailed study of real twisted K-theory and T-duality, which are then applicable to orientifold string theories. Anticipated outcomes include a deeper understanding of the theory of topological materials and its connection to string theory, and well-motivated mathematics widely applicable to the physical sciences. This understanding paves the way for novel technological applications.Read moreRead less
Advances in Index Theory. The laws of nature are often expressed in terms of differential equations which, if 'elliptic', have an 'index' being the number of solutions minus the number of constraints imposed. The Atiyah-Singer Index Theorem gives a striking calculation of this 'index', and this project involves innovative extensions of this theory with novel applications.