Many polymer models are obtained by perturbing measures on paths such as the measure on simple symmetric random walk on d-dimensional integer lattice or the Wiener measure. In this talk I'll discuss some of the properties of the typical paths under these polymer measures.

In 1980, A. P. Calderon published a short seminal paper entitled "On an inverse boundary value problem", which has become the starting point in the mathematical analysis of the following inverse problem: Can one determine the electrical conductivity of a medium by making current and voltage measurements at the boundary of the medium? To this day, this problem serves as a fundamental source of motivation and inspiration for many developments in the field of inverse boundary problems. In this talk we shall give an introduction to the field of inverse boundary problems, survey some of the most important developments, and state some open problems.