Focus Areas

Application-Driven Statistical Advancements

The advancement of academic statistics requires novel contributions to statistics in terms of new methodology. Such novelty often arises as a need for analytics of a real-world application presenting complexity not seen before. This area of statistical science aims to provide novel approaches to problems in the real world that are difficult for existing statistical frameworks to solve.

This interdisciplinary approach combines theoretical rigor with practical applicability, aiming to create impactful solutions across various domains. Developing software packages and tutorials that connect theoretical concepts with practical applications could be valuable. The application areas are diverse and could include spatial data, epidemiology, financial data, temporal data, health data and environmental data for example.

Researchers

Graph Theory

Graph Theory is a modern area of mathematics whose explosive growth in the past 70 years can be attributed to its wide range of applications, foremost in computer science and information technology. This focus area is at the heart of an innovation, which has promoted the information society arguably more than any other innovation: the search engine Google, whose success is a result of a novel use of graph theoretic methods. Graph Theory is also important for the development of algorithms for the routing of cellphone calls, which are vital for the functioning of communication in the South African society.

As a research area, Graph Theory has been very active. Several South African universities have research groups or individuals engaged in research in graph theory and closely related areas, including Stellenbosch University, University of Johannesburg, University of the Witwatersrand, University of Cape Town, University of the Western Cape, University of the Free State, Rhodes University, University of KwaZulu-Natal, and the University of Pretoria.

Researchers

Mathematical and Statistical Modelling in Biosciences

This focus area is devoted to recent advances in life sciences that are based on applications of mathematics and statistics. It also includes mathematics and statistics which is applied or motivated by biological studies. The focus area seeks to promote multidisciplinary research, bringing together mathematicians, applied mathematicians, statisticians, biologists, ecologists and other practitioners in life and medical sciences. With so many sub-themes, multidisciplinary research is at the forefront of the activities.

South Africa, like many other developing countries, is deeply affected by diseases such as HIV/AIDS, tuberculosis, malaria, and cancer, among others. Human health, the health of domestic and wild animals, plant life, and the broader environment are interconnected. Consequently, a knowledge-driven approach is essential to address the challenges at the human-animal-ecosystem-environment interface. The focus area calls for integration of various fields and disciplines, amongst others mathematics, statistics, ecology, biology, to collaboratively enhance well-being and address threats to health and ecosystems through sophisticated modelling, together with the computational aspects. Advancements in modelling methodologies may assist decision-makers in executing essential management judgments about these matters. 

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Mathematical Physics

Questions arising from Physics have always provided strong motivation for research in Mathematics, often leading to new techniques or even the emergence of new mathematical fields. Conversely, the detailed study of physical systems (including the Universe as a whole) very often requires advanced mathematical tools. Being at the interface of these two worlds, mathematical physics is fertile ground for identifying new directions in the mathematical sciences as well as for applying these techniques to new and useful settings.

The mathematical physics focus area encompasses a wide range of topics such as theoretical high energy physics, gravitational physics, theoretical cosmology, computational physics and integrable systems. They are all connected using mathematical techniques to aid in studying physical questions. Many of these techniques are motivated by recent developments in string theory, which have led to completely new ways of approaching the study of physical systems.

An important aspect of the focus area is that it provides perhaps one of the best avenues to popularise the importance of the mathematical sciences to the general public. As mentioned, it is often the deep questions about nature that provide the starting point for people of all ages to become interested in science. So, even though the message that mathematics underlies all aspects of daily life and technology should always be emphasised, the role of mathematics as the language of nature is an important element in engaging the public and perhaps breaking down some of the barriers when it comes to understanding and using mathematical concepts.

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Mathematical Sciences Education Research

Education is challenged to come up with new ideas about teaching and learning in the mathematical sciences, such as mathematics and statistics, where disciplines have to react to a fast-changing world.  Educators are challenged to use new curricula, devise new teaching strategies, and come up with new assessment methods that are in tune with the working world that graduates will be entering post-study. Accordingly, there is enormous pressure on and within university systems to improve many aspects of teaching and learning in the sciences and in mathematics and statistics in particular, with no shortage of ideas as to how professional practice could be positively developed.

In statistics and mathematics, however, there are additional factors that call for research and international collaboration to ensure best practices are followed in the education of these disciplines. Studies in Science, Commerce and Engineering, require at least one module to be offered by Mathematics or Statistics departments. This puts constraints on these disciplines, having to deal with large classes of mixed ability students.

The digital age has resulted in changes in the ways in which data is captured and analysed, so that statistics education needs to constantly be reviewed and updated to be in touch with the changing world that the graduates will be entering. There is further an exponential growth in the need for data analytics capabilities of non-statisticians, calling for statistics departments to run training workshops in statistical analysis, reporting and decision making from data for the University, Government, Banks and Industry. Statistics departments must react to a continuously changing work environment, produce “job ready” graduates and find ways to push the boundaries of the discipline and come up with innovative ideas in teaching.

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Methodological and Computational Statistics

An essential component and prerequisite for conducting good research is statistical theory and methodology. In a more theoretical sense, postulating methodological frameworks for complex data remains of certain value by ensuring continuous development and enhancement of theory-based probabilistic models that may serve the said phenomena of random behaviour mathematically. In this focus area consideration and acknowledgment should be given to significant recent advancements in the statistical discipline.

Most significantly, processing power has increased rapidly; big data and more extensive multidimensional data systems can now be analysed. The science of statistics focuses on modelling and learning from data in both theoretical and applied settings. A more applied context calls these theoretical developments to action, with approaches many times requiring computationally challenging environments in which the modelling effort may be enriched with subsequent estimation and hypothesis testing. This focus area intends to unify fundamental methodological research in statistics together with computational aspects of the modern era.

Researchers

Number Theory

The Number Theory Focus Area covers a wide range of topics in number theory and its related fields such as algebraic geometry, combinatorics, discrete mathematics, and model theory. Although, number theory primarily deals with the properties and relations of the whole numbers, it has become an exciting interdisciplinary field in mathematics with numerous applications in other areas of science such as in computer science and physics.

The role of the focus area is to connect South African researchers working in number theory and its related fields, as well as to facilitate international networking for the members of this focus area.

Number theory is one of the oldest and deepest branch of mathematics. Problems in number theory are often very simple to state, and this attracts mathematician of all ages to work in this field. As a result of this, many of the world’s brightest minds have considered problems in number theory. As we all aware, mathematics is crucial for the development of science and technology, and having number theory as a major research topic in South Africa will inspire young students in the country to consider a career in mathematics. Moreover, several methods developed in number theory have stimulated advances in other fields, such as computer science.

Researchers

Numerical and Applied Mathematics

This focus area strives to bring together various groups scattered throughout South Africa. Some of the larger groups are the Numerical Analysis and Scientific Computing group at Stellenbosch University, and the Fluid Dynamics and Modelling group at the University of the Witwatersrand, but there are also several other smaller groups and individual researchers. The common thread among these groups is computation and application. Another aim of the focus area is to introduce young learners to ideas of applied and numerical mathematics and encourage a future generation of researchers in these areas.

The major mechanism for improving countrywide collaboration is through an annual meeting. This is the annual South African Numerical and Applied Mathematics (SANUM) symposium. This symposium has acquired an international status since its inception in 1975, and attracts many plenary and contributed talks from well-known international researchers. At this meeting, applied mathematicians discuss their work and often pose computational challenges to the numerical mathematicians. In turn, numerical mathematicians advertise their skills on the latest computational challenges. In this way, collaborations to the mutual benefit of both sets of researchers can be established.

Researchers

Operator Algebras and Functional Analysis

The field of Operator Algebras and Functional Analysis is a very broad field drawing together a group of people with somewhat diverse interests who all share one common focus: the application of analysis in describing, understanding, and effectively applying mathematical structures. The most fruitful subfields of this area with the richest structure and most challenging problems are those which draw together a variety of structures into one integrated whole.

Functional analysis and operator theory holds an important position in modern mathematics, and many subjects have their foundations built on it; to mention just a few: quantum mechanics, partial differential equations, and numerical analysis, which is essential in all engineering applications.

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Statistical and Machine Learning

Statistical and Machine Learning (S&ML) combines traditional statistical methods with modern computational algorithms to analyse and model data. Statistical learning focuses on estimation and inference, using probabilistic models and assumptions to explain relationships between variables, while machine learning emphasises predictive accuracy through flexible, data-driven approaches. Classical methods such as regression, time series analysis, and hypothesis testing form the foundation of statistical learning. In contrast, machine learning techniques, including decision trees, neural networks, and ensemble methods, leverage computational power to handle complex, high-dimensional data.

The synergy between statistics and machine learning is evident in various contexts, such as regularisation (Ridge, LASSO, MEnet), Bayesian inference, and ensemble learning, which merge interpretability with predictive strength. While statistical methods remain crucial for hypothesis-driven analysis and explainability, machine learning excels in high-dimensional, unstructured environments where traditional models struggle. Integrating statistical rigor with machine learning’s adaptability enhances decision-making as data science advances, making S&ML essential in fields like finance, healthcare, and social sciences.

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Symmetries, Mechanics, and Applications

In this focus area, the notion of symmetry as well as its natural extension, which is of critical importance to modern science, plays a role.  The fundamental nature of symmetry ensures that it arises in several areas in mathematics, engineering and physics and therefore provides an arena for interdisciplinary research. 

This focus area is intended to be broad,and covers areas of research where theory and analytic methods associated with symmetry play a role, including mechanics, fluids, gravitational theories, mathematical and theoretical physics, dynamical systems, differential geometry and all types of differential and difference equations.

The aim of the focus area and associated sub-areas of investigation could provide avenues to broaden the interdisciplinary research across other focus areas.  Noting that this focus area is itself very broad, we strive on creating intersecting collaborative work.  These could be both applied as well as theoretical in nature. 

Researchers