For over a hundred years, when confronted by swelling in the brain, surgeons more often than not have resorted to decompressive craniectomy, the traditional route to reducing swelling by removing a large part of the skull. However, while this might be the standard procedure, its failure rate has been worryingly high, primarily because the consequences on the rest of the brain have been poorly understood. 
 

There is a wide class of problems in mathematics known as inverse problems. Rather than starting with a mathematical model and analysing its properties, mathematicians start with a set of properties and try to obtain mathematical models which display them. For example, in mathematical chemistry researchers try to construct chemical reaction systems that have certain predefined behaviours. From a mathematical point of view, this can be used to create simplified chemical systems that can be used as test problems for different mathematical fields.

Correctly predicting extinction is critical to ecology. Claim extinction too late, and you may be taking resources away from a species that actually could be saved. Claim extinction too early, and you may cause the true extinction due to stopping resources, such as removing protection of its habitat.

Numerous processes across both the physical and biological sciences are driven by diffusion, for example transport of proteins within living cells, and some drug delivery mechanisms. Diffusion is an unguided process which is of great importance at small spatial scales.