Abstract:
In many models of physical and biological processes, interesting spatio-temporal dynamics occur in certain parameter regimes whereby the solution is highly localized at certain discrete points of the domain. Such features, known as spikes in 1D and spots in 2D, can undergo splitting, replication and oscillatory instabilities. Alternatively, the domain itself may be perturbed by a discrete number of localized topological defects such as holes. We refer generally to such features as strongly localized phenomena. For these problems, specialized singular perturbation techniques are available to exploit the discrete nature of these phenomenon and determine finite dimensional reductions of the infinite dimensional problem which retain its essential features.

In this talk I will outline these general techniques and discuss three examples where they can be applied. The first example comes from an Ecological setting where the goal is to find the optimal configurations of localized resources patches in a habitat. In the second example, the problem is to determine the perturbations to the spectrum of certain linear eigenvalue problems due to a configuration of holes or defects. The third problem is an inverse problem whereby the goal is to find the location and size of a finite number of inhomogeneities in the interior of a region, given only the boundary data.