Polymers are prototypical examples of soft-matter systems. The talk will first focus on equilibrium statistical mechanics, and introduce basic concepts like the random walk and the self-avoiding walk. This is complemented by a discussion of the notion of coarse-graining and scale invariance, which is at the basis of modeling polymers in terms of simple bead-spring models. The second part will then discuss the basics of polymer dynamics, in terms of the fundamental Rouse, Zimm, and reptation models. The third part is devoted to a brief overview over Monte Carlo and Molecular Dynamics models and simulation algorithms, which are directly based upon the insight into the essential physics. If time permits, a brief outlook on the physics of membranes will be added.

Joint work with Eric Vanden-Eijnden.

In many problems of multiscale modeling, we are interested in capturing the macroscale behavior of the system with the help of some accurate microscale models, bypassing the need of using empirical macroscale models. This paper gives an overview of the recent efforts on establishing general strategies for designing such algorithms. After reviewing some important classical examples, the Car-Parrinello molecular dynamics, the quasicontinuum method for modeling the deformation of solids and the kinetic schemes for gas dynamics, we discuss three attempts that have been made for designing general strategies: Brandt's renormalization multi-grid method (RMG), the heterogeneous multiscale method (HMM) and the "equation-free" approach. We will discuss the relative merits and difficulties with each strategy and we will make an attempt to clarify their similarities and differences. We will then discuss a general strategy for developing seamless multiscale methods for this kind of problems. We will end with a discussion of the applications to free energy calculations and a summary of the challenges that remain in this area

Beginning with some observations about the periodic and nonperiodic structures commonly adopted by elements in the periodic table, I will introduce a definition ("objective structures") of a mathematically small but physically well represented class of molecular structures. This definition will be seen to have an intimate relation to the invariance of the equations of quantum mechanics. The resulting framework can be used to design various multiscale methods, and gives a new perspective on some of the fundamental solutions in continuum mechanics for solids and fluids. Open mathematical problems will be highlighted.