In contrast to the scalar-valued situation, solutions of vector-valued variational problems, more generally solutions of elliptic systems, can exhibit singularities, even when the system is "nice" (for example, uniformly elliptic with smooth coefficients), and even when the solution is "nice" (for example, a minimum of a variational integral). This motivates the study of the partial regularity of such systems. Typical questions to consider are:

• How large can the singular set of a solution be?
• Can one find conditions (on the coefficients of the system, or on the type of solution being considered) which guarantee that a solution is in fact everywhere regular?

After explaining the setting, we discuss a number of results in this area; some old, some recent.

Convenor: Simon Clarke.