Research_

Computation and algorithms

Overcome obstacles for efficient, secure and scalable computation

We build a mathematically sound foundation for the design of modern computational thinking and develop analytic frameworks for their correctness, performance, reliability and security.

The prevailing trend in mathematical research towards algorithmic and constructive processes is one of long-term importance.

Members of the Computation and Algorithms group investigate mathematical underpinnings of computing, with the goal of designing algorithms and computational approaches to problems in engineering, the social and natural sciences, medicine and healthcare.

The overarching goal of the group is to build a mathematically sound foundation for the design of modern computational thinking and to develop analytic frameworks for their correctness, performance, reliability and security.

The Computation and Algorithms group benefits from a unique environment where all branches of Mathematics and Statistics come together.

Members of the group uncover the fundamental abilities and limitations of the computational tools they work with, characterise the complexity of computational problems and provide innovative solutions for the design of efficient, secure and scalable numerical methods and algorithms for real-world applications.

*Header image: Simon Luo: Blind source separation for the Cocktail party*

Key researchers: Georg Gottwald, Lindon Roberts, Geordie Williamson, Dinxuan Zhou, Pengyi Yang.

In the past decade, machine learning has led to a paradigm shift in computer vision and language processing with remarkable success in, amongst others, medical image analysis and drug discovery.

Mathematics has an integral part to play and is in the unique position to unravel some of the mysteries of the success of machine learning algorithms, drawing from numerous areas within mathematics including numerical analysis, optimization, probability theory, group theory, approximation theory and dynamical systems.

We develop and apply machine learning algorithms to formulate conjectures in pure mathematics with the aim to prove them. We apply machine learning to understand molecular trans-regulatory networks in complex systems biology and develop new methods to forecast dynamical systems.

Key researchers: Eduardo Altmann, Jennifer Chan, Georg Gottwald, Clara Grazian, Uri Keich, John Ormerod, Ellis Patrick, Michael Stewart, Garth Tarr, Rachel Wang, Geordie Williamson, Jean Yang, Pengyi Yang.

The advent of Big Data has revolutionised science and industry, bringing new challenges and opportunities in the form of large-scale, heterogeneous, and dynamical data.

Much of the research in computational statistics and machine learning is inspired by practical problems in data-rich disciplines including bioinformatics, artificial intelligence, finance, and social science, where mathematical and algorithmic creativity are interwoven into statistical methodologies.

Our research spans a diverse spectrum from theory to applications. We tackle challenging theoretical problems concerning the mathematical and statistical foundations of machine learning frameworks and algorithms.

We develop innovative and efficient computational methods to extract insights from data, building predictive models and performing statistical inference.

Key researchers:

Geoff Bailey, John Cannon, Allan Steel, Nicole Sutherland, Don Taylor, Bill Unger, John Voight, Geordie Williamson

Computers allow us to explore and solve hard problems in pure mathematics. In computational algebra, we study a wide range of topics--including algebra, number theory, geometry, representation theory, and combinatorics--through the lens of symbolic computation and with an eye to explicit structures.

Using sophisticated algorithms, we can manipulate complex mathematical objects, provide examples or counterexamples, and verify statements that would be otherwise challenging or impossible to handle.

Our work is focused on the development of Magma, a large, well-supported software system designed for algebraic computation.

Magma was first released in 1993 at the University of Sydney and has received contributions from hundreds of mathematicians worldwide.

Magma provides a mathematically rigorous environment, a vast library of algorithms, and a flexible language for defining and working with many structures in pure mathematics, including groups, rings, fields, modules, algebras, schemes, curves, graphs, designs, codes, and more.

- Computational and numerical mathematics