Calculating infinity minus infinity

Establishing stability and instability of stationary states is a basic requirement for understanding dynamics. Counting the dimension of the unstable directions is widely used as an (in)stability index. When applying these dynamical systems ideas to spatio-temporal pattern formation or to questions in geometry, one needs to expand the finite dimensional picture to an infinite dimensional setting. Working in an infinite dimensional space leads to difficulties: some infinite dimensional objects are larger than others. Fortunately, a well-established ingenious theory says that the difference in size between two such infinite dimensional objects can be measured by a relative index, essentially giving meaning to "infinity minus infinity". But how do you count the difference between two infinities in practice? In a paper published in the journal Foundations of Computational Mathematics, VU mathematicians Jan Bouwe van den Berg and Rob van der Vorst collaborated with Jean-Philippe Lessard (McGill, Canada) and Marcio Gameiro (Rutgers, USA) to develop a method to calculate such relative indices based on computer-assisted proof methods. As an example application, the computations lead to results about reaction-diffusion waves of propagating patterns in cylinders.