Daniel Gomez, University of Pennsylvania: How long will a confined Brownian particle take to hit an exceedingly small target? It is a classical result that the expected value of this first hitting time (FHT) blows up as the size of the target vanishes in two or more spatial dimensions. This is an example of a "strongly localized perturbation'' in the sense that small geometric defects have large global effects. If Brownian motion is replaced with Lévy flights, a spatially discontinuous jump process, then the FHT has the potential to blow up even in the case of one spatial dimension. In this talk, I will discuss how matched asymptotic expansions yield a computationally inexpensive method for computing the FHT in the case of Lévy flights by reducing the problem to that of solving a linear system of equations. Moreover, we will see that depending on the fractional order of the Lévy flight, the FHT is qualitative similar to that for Brownian motion in one or more spatial dimensions. In addition to analyzing FHT problems, matched asymptotic expansions have also been highly successful in studying localized solutions to singularly perturbed reaction diffusion systems. I will conclude by outlining how matched asymptotic expansions similarly yield nonlinear algebraic systems, globally coupled eigenvalue problems, and differential algebraic equations that describe the structure and dynamical properties of localized solutions.