An optimisation-based framework for non-classical Chebyshev approximation. This project aims to solve open mathematical problems in multivariate and piecewise polynomial approximations, two directions that correspond to fundamental obstacles to extending classical approximation results. Through an innovative combination of optimisation and algebraic technique, the project intends to develop foundations for new results in approximation theory, and new insights into other areas of mathematics, mos ....An optimisation-based framework for non-classical Chebyshev approximation. This project aims to solve open mathematical problems in multivariate and piecewise polynomial approximations, two directions that correspond to fundamental obstacles to extending classical approximation results. Through an innovative combination of optimisation and algebraic technique, the project intends to develop foundations for new results in approximation theory, and new insights into other areas of mathematics, most notably optimisation. The techniques and methods developed should also have significant benefits in the many disciplines where approximation problems appear, such as engineering, physics or data mining. The research outputs resulting from this project will be used in a wide range of fields to help implement programs, policies and improve decision making.Read moreRead less
A new asymptotic toolbox for nonlinear discrete systems and particle chains. This project aims to pioneer a mathematical toolbox of new asymptotic techniques for discrete systems driven by vanishingly small influences. The purpose of these techniques is to permit the asymptotic study of discrete problems in which significant effects originate due to subtle causes that are invisible to existing asymptotic methods. Discrete systems play a significant role in modern applied mathematics, and it is v ....A new asymptotic toolbox for nonlinear discrete systems and particle chains. This project aims to pioneer a mathematical toolbox of new asymptotic techniques for discrete systems driven by vanishingly small influences. The purpose of these techniques is to permit the asymptotic study of discrete problems in which significant effects originate due to subtle causes that are invisible to existing asymptotic methods. Discrete systems play a significant role in modern applied mathematics, and it is vital that mathematical tools be designed in order to explore their behaviour. The aim of this project is to open new pathways for resolving open scientific problems, providing benefits such as understanding the energy dissipation of particle chains and granular lattices contained in small-scale technological components.Read moreRead less
Creating Hybrid Exponential Asymptotics for use with Computational Data. Asymptotic analysis is a vital tool for studying small influences with critical effects. This project aims to create an innovative fully-automated asymptotic framework for studying phenomena which are invisible to classical approximation methods, using new ideas from asymptotics and numerical complex analysis. The outcome will be the first framework that can be used on data from numerical simulations or real-life measuremen ....Creating Hybrid Exponential Asymptotics for use with Computational Data. Asymptotic analysis is a vital tool for studying small influences with critical effects. This project aims to create an innovative fully-automated asymptotic framework for studying phenomena which are invisible to classical approximation methods, using new ideas from asymptotics and numerical complex analysis. The outcome will be the first framework that can be used on data from numerical simulations or real-life measurements, and which can be applied automatically without hands-on expert input. It will be used to design submerged structures and efficient vessels with minimal energy loss from surface waves. Expected benefits include making powerful methods accessible to scientists, and new paths for energy-efficient industrial design.Read moreRead less
High Dimensional Approximation, Learning, and Uncertainty. This project aims to develop next-generation computational methods for complex problems in science and engineering that have many uncertain parameters, using advanced high-dimensional strategies and deep learning to enhance computational speed. The significance of the project is that these methods will help address important applications that at present are not feasible or at the edge of feasibility. The expected outcomes are powerful me ....High Dimensional Approximation, Learning, and Uncertainty. This project aims to develop next-generation computational methods for complex problems in science and engineering that have many uncertain parameters, using advanced high-dimensional strategies and deep learning to enhance computational speed. The significance of the project is that these methods will help address important applications that at present are not feasible or at the edge of feasibility. The expected outcomes are powerful methods that will be mathematically rigorous and suitable for a wide variety of applications. The benefits are that the project will boost Australia’s position as a leader in innovation, and contribute to future developments over a wide area, from aerospace engineering to personalised computational oncology.Read moreRead less
Approximation theory of structured neural networks . Mathematical theory for deep learning has been desired due to the power applications of deep neural networks to deal with big data in various practical domains. The main difficulty lies in the structures and architectures imposed to networks designed for specific learning tasks. Neither the classical approximation theory nor the recent one for depths of ReLU neural networks can be applied due to the structures imposed for processing large dime ....Approximation theory of structured neural networks . Mathematical theory for deep learning has been desired due to the power applications of deep neural networks to deal with big data in various practical domains. The main difficulty lies in the structures and architectures imposed to networks designed for specific learning tasks. Neither the classical approximation theory nor the recent one for depths of ReLU neural networks can be applied due to the structures imposed for processing large dimensional data such as natural images of tens of thousands of dimensions. This project aims at an approximation theory for structured neural networks. We plan to establish mathematical theories for deconvolution with deep convolutional neural networks, operator learning, and spectral graph networks. Read moreRead less
High Dimensional Computation and Uncertainty. This project aims to establish powerful computational methods for high-dimensional problems - methods that are rigorous, and carefully tailored to specific applications, from physics, environment, manufacturing and finance, and often driven by uncertainty. The project will generate new knowledge in the area of high-dimensional computation, and develop technological innovations in key areas of science and industry. Expected outcomes include improved c ....High Dimensional Computation and Uncertainty. This project aims to establish powerful computational methods for high-dimensional problems - methods that are rigorous, and carefully tailored to specific applications, from physics, environment, manufacturing and finance, and often driven by uncertainty. The project will generate new knowledge in the area of high-dimensional computation, and develop technological innovations in key areas of science and industry. Expected outcomes include improved control of uncertainty in industry, enhanced international and interdisciplinary collaborations, and significant publications and presentations in international forums. The technological advancements will help boost Australia's position as a world leader in innovation.Read moreRead less
Geometric analysis of nonlinear systems. Modern science derives its power from mathematics. The project aims to capture, identify and describe pivotal, transcendental solutions of nonlinear systems that are universal in science, in the sense that they always arise as mathematical models under certain physical limits. The project expects to produce new mathematical methods to describe such functions by using a newly discovered geometric framework. Expected outcomes include the description of elus ....Geometric analysis of nonlinear systems. Modern science derives its power from mathematics. The project aims to capture, identify and describe pivotal, transcendental solutions of nonlinear systems that are universal in science, in the sense that they always arise as mathematical models under certain physical limits. The project expects to produce new mathematical methods to describe such functions by using a newly discovered geometric framework. Expected outcomes include the description of elusive solutions of discrete and higher-dimensional nonlinear systems. This should provide significant benefits, such as new mathematical knowledge, innovative techniques, enhanced scientific capacity in Australia.Read moreRead less
Broadening Choice and Increasing Diversity in Public Schools. Currently, most families are limited to the public school in their catchment area, meaning the area in which they can afford to live. This leads to socio-economically and ethnically homogenous schools and entrenches disadvantage, as well as denying students the crucial life lessons that flow from being part of a diverse student body. This project aims to investigate a model for allocating public school places that integrates catchment ....Broadening Choice and Increasing Diversity in Public Schools. Currently, most families are limited to the public school in their catchment area, meaning the area in which they can afford to live. This leads to socio-economically and ethnically homogenous schools and entrenches disadvantage, as well as denying students the crucial life lessons that flow from being part of a diverse student body. This project aims to investigate a model for allocating public school places that integrates catchment areas. The expected outcome would be a system that gives families a wider choice, enabling them to enrol in out-of-area schools, while ensuring that allocations remain fair, equitable and balanced, and also delivering benefits such as achieving a desired level of diversity in student populations within schoolsRead moreRead less
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: 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