Mathematics – from unsolved equations to disease, climate and finance
In mathematics, our research spans a broad range of applied problems in society, while also delving deeply into issues within the discipline. Artificial intelligence (AI) in healthcare, financial risk management, the motion of water in waves and the element of chance are just a few examples.
Probability theory and its benefits for healthcare, the environment and finance
Mathematical statistics is one of our fields of research. The focus here is on the concepts of probability and chance. The theoretical research is aimed at probability theory, statistical theory and methodology. For example, random graphs, random walks and neural networks are studied, with close links to statistical physics and neuroscience.
Applied research in mathematical statistics is geared towards three specific areas. The first area relates to biology and medicine, where mathematical statistics contributes to bioinformatics research on analysing genetic links to hereditary diseases. Diagnostic methods and AI are also being developed to provide enhanced analysis of medical images and signals.
The second applied area of mathematical statistics concerns the environment, climate and risk. One key research question relates to how a slowly changing climate can lead to extreme weather events, and how historical climate can be reconstructed from measurements.
The third area focuses on finance and methods for various financial forecasts. We develop methods for financial risk management, optimisation and pricing, which have direct applications in the transition to a sustainable energy system.
Calculations solve a wide range of problems
Our research at the Centre for Mathematical Sciences also includes numerical analysis and computational techniques. Mathematical modelling describes a problem in terms of a mathematical equation, often a partial differential equation. The research field is concerned with developing methods for solving such complicated mathematical equations by computer.
In this field, our researchers work on designing, analysing and programming efficient solution algorithms that can help tackle a wide range of problems in industry, engineering and science. This may involve calculating, for example, planetary orbits, hip and knee joint prostheses, strength and cracking in different materials, complex flow problems in weather and climate forecasting, or creating digital twins of technical systems.
Studying symmetries
Algebra is another of our research areas. In this case, our researchers work extensively with group theory, which involves studying symmetries. A group is simply a collection of symmetries, such as the rotation of a cube around an axis. Group theory has far-reaching applications not only in other branches of mathematics but also in other sciences, such as physics, chemistry, computer science and engineering. Our researchers work with infinite groups and analyse their structure.
Four different approaches to mathematical analysis
Mathematical analysis is another of our research areas. Basically, this is about changes, especially different kinds of limits such as derivatives and integrals. This research is not only purely theoretical but is also linked to applications in fluid dynamics, statistical physics, control engineering, optimisation and inverse problems, including ties with the MAX IV Laboratory.
Our researchers in mathematical analysis work with four different approaches. Using partial differential equations we develop a theoretical basis for modelling complex phenomena such as water waves or gas flows. Harmonic analysis allows us to break down complicated behaviours into simpler periodic patterns. In the case of operator theory, we use a powerful framework to study general classes of linear maps. Lastly, we use complex analysis, which concerns the surprisingly rigid properties of complexly derivable functions and is fundamental to many applications in science and engineering.
Possibilities offered by differential geometry
Differential geometry is also one of our fields of research. The subject involves abstract differentiable manifolds, which are a generalisation of more familiar geometric objects such as curves and surfaces. This field has many key applications and is, for example, fundamental to Einstein’s general theory of relativity. At Lund University harmonic morphisms between Riemannian manifolds are studied, which can be seen as a generalisation of complex analysis.
