A liquid crystal (lc) is, roughly, a fluid equipped with a field of directions (think of the local direction of an oblong molecule), that is: a mapping into the unit sphere S. Associated with this mapping the fluid exhibits an energy density, which measures the local nonuniformness of the "director field". If no flow is present, an equilibrium situation is typically a stationary point of the corresponding energy integral over the region R, occupied by the lc. This establishes a connection with the theory of harmonic maps with the unit sphere as target. We discuss the situation where R is an infinitely long cylinder and where the situation is cylindrically symmetric. So, R may be replaced by the unit disk D. In this situation, we are confronted with the following key question:

Given a sufficiently smooth map v: D --> S, is there a harmonic map u, homotopic with v ?

Eells and Sampson proposed to study this question through the associated "heat flow", which is the parabolic problem corresponding to the (elliptic) harmonic map equation. A central theme of the lecture will be the blow-up behavior of this harmonic map heat flow. Regretfully, this heat flow problem is too simple to describe the dynamical behavior of an lc. We shall describe the character of the additional problems. Most of this is joint work with Michiel Bertsch, Roberta Dal-Passo and Elisabetta Vilucchi (Roma II).