Can maths tell what happened?
Prof. Chris Budd, Department of Mathematical Sciences, University of Bath
On arrival at a crime scene or at an accident, the police may be faced with a set of evidence, but no clear indication of the precise cause of this evidence. For example, a speeding motorist may leave a set of tyre skid marks, but can we tell from these how fast they were going before an accident occurred? One way to obtain an answer to this question is to use the method of mathematical modelling. In this, an agreed model is determined for the event under investigation which allows a link to be made between a cause and its effects. From this a possible set of causes consistent with the evidence can be found, and from this we can learn more about what happened. In the case of the speeding motorist we can find an estimate for the speed, provided that we know the other factors involved, such as the frictional force between the tyres and the road. This process of finding a mathematical model for an event and then linking cause to effect is called solving an inverse problemand is an important modern application of mathematics. It is very like trying to work out the shape of an object knowing what shadows it casts. Very similar techniques are also used in medical imaging, oil prospecting and remote sensing by satellites. Mathematics gives a means for rapidly transferring ideas from one application to another!
In my presentation I will be looking at five different applications of this method to problems involving crime detection. In addition to the speeding motorist these will be: an investigation of finding where a contaminant was released into a water supply, deblurring a blurred photograph of a number plate, finding out what or who killed King Tutankhamen and (in helping to solve a crime against humanity) a method for detecting anti-personel land-mines. Remarkably the maths which is used to find the land-mines is closely related to a method for finding brain tumours in a CAT scan and has its origins in a formula discovered by the mathematician Radon at the beginning of the 20th Century!
Mathematics is also used in such diverse areas as fingerprint recognition, finding the trajectories of bullets and assessing the reliability of evidence such as marks left by tools. I should emphasise that the use of mathematics won’t (nor should it) solve every problem, and it has its limitations like any other procedure, but it a useful tool in the set of techniques used in the forensic service.
My objective in this presentation is to show that mathematics is important and highly relevant to crime fighting in particular, and to many other real life problems in general. Hopefully this will give an answer to the often asked question what’s the use of mathematics?
Most of the work I will present will be a summary of techniques used by others. However, the University of Bath is very active in applying mathematics to many problems arising in industry and medicine, ranging from micro-wave cooking to the design of aircraft structures and from studies of epidemics to complex systems. It is remarkable how often ideas which might be thought of as pure mathematics often find very real and important applications. The newly formed Bath Institute for Complex Systems exists to promote links between mathematics and many other disciplines and is always on the look out for new applications and challenges.