Geometry and analysis of discrete integrable systems. Whether we are looking at waves at a beach or the movement of herds of animals in a landscape, their movements and fluctuations turn out to rely on rules expressed by non-linear systems of mathematical equations. The aim of this project is to create a new mathematical theory to describe and predict the solutions of such systems.
Discrete Projective Differential Geometry: Comprehensive Theory and Integrable Structure. Differential geometry has been developed over centuries by the most distinguished of mathematicians and its applicability in the mathematical and physical sciences is beyond doubt. However, both natural and man-made structures are inherently discrete. Discrete differential geometry constitutes a relatively new and active research area located between pure and applied mathematics which is more fundamental th ....Discrete Projective Differential Geometry: Comprehensive Theory and Integrable Structure. Differential geometry has been developed over centuries by the most distinguished of mathematicians and its applicability in the mathematical and physical sciences is beyond doubt. However, both natural and man-made structures are inherently discrete. Discrete differential geometry constitutes a relatively new and active research area located between pure and applied mathematics which is more fundamental than differential geometry in that it aims to establish an autonomous discrete analogue from which differential geometry may be derived via an appropriate continuum limit. Even though discrete differential geometry has reached a high degree of sophistication, this project seeks to deliver the first comprehensive theory in this area. Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE140101366
Funder
Australian Research Council
Funding Amount
$376,527.00
Summary
Fully nonlinear partial differential equations in optimisation and applications. Fully nonlinear partial differential equations of Monge-Ampere type and their applications in optimal transportation have been studied intensively in the past two decades. Optimal transportation is a subject in linear optimisation. This project will develop a new theory on Monge-Ampere type equations in nonlinear optimisation, which is a much broader area with many significant applications. This project will investi ....Fully nonlinear partial differential equations in optimisation and applications. Fully nonlinear partial differential equations of Monge-Ampere type and their applications in optimal transportation have been studied intensively in the past two decades. Optimal transportation is a subject in linear optimisation. This project will develop a new theory on Monge-Ampere type equations in nonlinear optimisation, which is a much broader area with many significant applications. This project will investigate fundamental properties of solutions to this problem and explore further real-world applications. The outcomes of this project will have a substantial impact on partial differential equations and related research subjects. Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE180100957
Funder
Australian Research Council
Funding Amount
$339,328.00
Summary
Partial differential equations, free boundaries and applications. This project aims to investigate fundamental problems in the analysis of partial differential equations and free boundary theory, to develop advanced mathematical theories with the possibility of important applications. The expected outcome is the establishment of a regularity and classification theory for nonlocal equations and for free boundary problems in linear and nonlinear settings. The benefit of the project lies in a concr ....Partial differential equations, free boundaries and applications. This project aims to investigate fundamental problems in the analysis of partial differential equations and free boundary theory, to develop advanced mathematical theories with the possibility of important applications. The expected outcome is the establishment of a regularity and classification theory for nonlocal equations and for free boundary problems in linear and nonlinear settings. The benefit of the project lies in a concrete advancement of the mathematical research with advantages for a deeper understanding of complex phenomena in physics and biology. Some of the problems also provide results useful for industrial applications.Read moreRead less
Stochastic Geometry for Multi-sensor Data Fusion System. The aim of this project is to develop efficient algorithms for tracking and sensor management in a multi-sensor multi-target environment. Finite random set theory provides a natural way of representing a random number of (random) object states, an issue that has been largely ignored in the tracking literature until recently. Although a satisfactory foundation for multiple object filtering has been provided by random set theory, in this ear ....Stochastic Geometry for Multi-sensor Data Fusion System. The aim of this project is to develop efficient algorithms for tracking and sensor management in a multi-sensor multi-target environment. Finite random set theory provides a natural way of representing a random number of (random) object states, an issue that has been largely ignored in the tracking literature until recently. Although a satisfactory foundation for multiple object filtering has been provided by random set theory, in this early stage no algorithm capable of tracking many targets has emerged from this framework. We are confident that efficient algorithms can be developed by exploiting the insights and mathematical tools of stochastic geometryRead moreRead less
Discrete differential geometry: theory and applications. Sophisticated freeform structures made of glass and metal panels are omnipresent and their architectural design has been shown to be intimately related to a new area of mathematics, namely discrete differential geometry. This project is concerned with the theoretical basis of discrete differential geometry and its real world applications.
From quantum integrable systems to algebraic geometry and combinatorics. The purpose of this project is to investigate the deep connections that have recently emerged between the study of an area of mathematical physics (quantum integrable systems) and subjects of pure mathematics (enumerative and algebraic combinatorics, and algebraic geometry). These connections have a common root, which this project plans to reveal using novel methods coming from quantum integrability. This approach is expect ....From quantum integrable systems to algebraic geometry and combinatorics. The purpose of this project is to investigate the deep connections that have recently emerged between the study of an area of mathematical physics (quantum integrable systems) and subjects of pure mathematics (enumerative and algebraic combinatorics, and algebraic geometry). These connections have a common root, which this project plans to reveal using novel methods coming from quantum integrability. This approach is expected to illuminate these subjects leading to a new unified and interdisciplinary picture, and to resolve important open problems in the study of certain algebraic varieties and of their cohomology in the theory of symmetric functions, and related counting problems.Read moreRead less
Tantalizer algebras and generalized lattice models. This project exploits underlying symmetry to characterise components and flow patterns in network configurations. The project will develop tools for analysis and optimisation of systems of interacting nodes which can arise in materials, computing networks, and any social or industrial contexts with communication or product transfer between nodes.
Analysis of nonlinear partial differential equations describing singular phenomena. This project will advance knowledge on a huge variety of systems with applications across the sciences by providing new methods to investigate nonlinear partial differential equations with singularities. The analysis of many models describing physical and biological systems relies on such equations.
Multivariate polynomials: combinatorics and applications. Many combinatorial and computational problems in pure and applied mathematics as well as mathematical physics can be solved using polynomials in many variables. This proposal will expand our understanding of the poorly understood class of non-symmetric polynomials by studying their novel combinatorial structure. The outcomes will address the current difficulty of implementing non-symmetric polynomials in symbolic algebra packages by devel ....Multivariate polynomials: combinatorics and applications. Many combinatorial and computational problems in pure and applied mathematics as well as mathematical physics can be solved using polynomials in many variables. This proposal will expand our understanding of the poorly understood class of non-symmetric polynomials by studying their novel combinatorial structure. The outcomes will address the current difficulty of implementing non-symmetric polynomials in symbolic algebra packages by developing completely new computational algorithms. Secondly, this new understanding will be used to solve several challenging mathematical enumeration problems.Read moreRead less