In many areas of applied sciences, engineering and technology there are three problems dealing with data and signals: (i) data compression; (ii) signal representations; and (iii) recovery of signals from partial or indirect information about the signals, often contaminated by noise. Major advances in these problems have been achieved in recent years where wavelets, multiresolution analysis, and kernel methods have played key roles. We consider problem (iii) and give an overview of specific contributions to inverse and ill-posed problems where reproducing kernel Hilbert spaces provide a natural setting.

There is a rich physical literature on polymer dynamics which presents a number of fascinating challenges for mathematicians. We model thermal fluctuation of a polymer in solvent as a curve or loop obeying a stochastic partial differential equation (SPDE). The simplest instance is the so-called Rouse model which is an infinite dimensional Ornstein-Uhlenbeck process satisfying a linear SPDE. We'll review the Rouse model and then describe recent results (a) of Seung Lee on an SPDE for the Rouse model in a half-space with reflecting boundary conditions; and (b) of Scott McKinley on an SPDE model of the hydrodynamic interaction.