We will talk about aperiodic tilings (such as Penrose tilings) and quasicrystals, and will see how questions on spectral and quantum dynamical properties of quasicrystals lead to questions on Cantor sets, singular measures, and hyperbolic dynamical systems.
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.