Mathematical modelling is a powerful tool for understanding, describing and predicting responses in real-world systems. When it is done well, it goes far beyond quantitative predictions and gives real insight into how and why different inputs affect different outputs. In my research group, we use it primarily for studying physical systems, but it can also be used for societal problems, chemistry and biology problems, and beyond.
Areas that I am particularly interested in using mathematical modelling for include:
Wrinkles: a thin sheet will buckle and wrinkle when compressed, but why? What determines the wrinkle pattern, and can we control it?
Manufacture: carefully-tuned manufacture processes are required for making thin sheets of both metal and glass, for applications ranging from smartphone screens to car bodies. These processes are not always perfect and are under constant improvement, and we use mathematical modelling to help with this.
Decontamination: when a toxic chemical spills, how do you clean it up? What if it soaks into a porous material, such as a concrete floor? Experiments are difficult because they involve dangerous chemicals and because it is difficult to visualize what is happening inside a concrete slab, so mathematical modelling is an invaluable tool for testing scenarios safely.