Research_

Data and modelling

Detecting complex patterns in systems and data sets

Developing computational and statistical methods for real-world systems.

The construction of mathematical models in the sciences, medicine, technology and finance is an important and rapidly expanding area of research. Members of our Data & Modelling research cluster thus work in close collaboration with interdisciplinary partners and experimentalists.

Computational Mathematics, Mathematical Modelling, Statistical Analysis and Data Science occupy an important and growing role. Our researchers design efficient algorithms for processing large amounts of data, develop dynamical models to explain natural phenomena and discover new ways to detect and interpret complex patterns in real-world systems.

*Header image: Eduardo Altmann: Hierarchical clustering of Wikipedia articles*

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: Chris Bertram, Daniel Daners, Peter Kim, Robert Marangell, Mary Myerscough, Martin Wechselberger.

We use mathematics to explore fascinating and complex questions of living systems. To do so, we team up with collaborators from relevant disciplines, creatively build mathematical models, and seek to provide insights from mathematical reasoning.

Our mathematical biology group works in areas as diverse as anthropology, collective behaviour, ecology, epidemiology, immunology, medicine, neuroscience, physiology, social insects, and beyond.

These systems span across a vast breadth of spatial scales, from the microscale study of bacteria and viruses to the global scale of human populations and ecosystems. Timescales range from milliseconds for neural transmission to millions of years for evolutionary adaptation.

Our mathematical biologists use mathematical, statistical, and computational tools to develop, improve and solve mathematical models. We use differential equations, difference equations, and agent-based models to capture the dynamics of the world around and within us.

Key researchers: Jean Yang, John Ormerod, Uri Keich, Pengyi Yang, Ellis Patrick, Rachel Wang, Garth Tarr, Clara Grazian, Shila Ghazanfar.

Bioinformatics is an interdisciplinary field utilising quantitative methods from computer science, mathematics, and statistics to analyse and interpret large biological datasets. We develop and apply methods for applications ranging from the identification of predictive biomarkers of disease to characterising molecular signalling patterns within cells.

We share an interest in developing statistical and computational methodologies to tackle the foremost significant challenges posed by modern biology and medicine.

Most of the researchers in the School of Mathematics and Statistics working in Bioinformatics are also members of the Bioinformatics Cluster in the Sydney Precision Data Science Centre.

Some of the software our group develop can be found on the public repository of Sydney Precision Bioinformatics on GitHub.

**Seminars**

Key researchers: John Ormerod*, *Jennifer Chan, Clara Grazian, Rachel Wang.

Bayesian statistics is an approach to data analysis where epistemological uncertainty associated with model parameters is described using probability distributions. As data is gathered, Bayes Theorem is used to update the probability distribution of model parameters given the observed data.

Bayesian methodology is used in diverse fields such as machine learning, medicine, ecology, insurance, finance and astronomy, to name just a few.

Research in Bayesian statistics at the University of Sydney is at the forefront of the area. Our group conducts research in the Bayesian analysis of time series data, including stochastic volatility models, modelling of cryptocurrencies via multivariate time series, and modelling of COVID-19 case numbers.

We consider approximate Bayesian inference methods where numerical accuracy is sacrificed in order to enable methods for complex datasets arising in machine learning and bioinformatics.

Finally, we consider problems involving various forms of clustering (including mixture modelling), nonparametric Bayesian methods, and community detection methods on graphs.

- Econometrics and financial mathematics
- Fluid dynamics