In model theory, we study the definable sets in a structure.This becomes applicabe to another field of mathematics ifthe definable sets are the objects of study in that field.For example, the definable sets in an algebraically closed field are precisely the constructible sets, and hence thetools of model theory can be used in algebraic geometry.Mathematically, one also studies quotients, but this canbe a problem model-theoretically, as the quotient by a definable equivalence relation in general cannot be expected to be definable. If every quotient can be identified with a definable set, we say that the structure eliminates imaginaries.Algebraically closed fields do eliminate imaginaries, but valued fields in general do not, at least in the simplest language for studying them. In this talk, I will explain the above ideas more precisely, discuss the obstacles to eliminatingimaginaries in valued fields, and describe the richer languagein which valued fields do eliminate imaginaries, as provedin recent work by myself, Ehud Hrushovski and Dugald Macpherson.

I will describe an existence theorem for traveling waves in water. Thisis aproblem of the dynamics of a free surface of an incompressible fluid.The first suchresult in two dimensional settings is due to T. Levi-Civita and D.Struik in the 1920's.In a recent paper we prove a general result for three dimensions (well,for anynumber of dimensions), when there is surface tension. The approach issurprisinglyclose to the Lyapunov center theorem of A. Weinstein, using the fact due toV. E. Zakharov that the water waves problem is a Hamiltonian system.Withoutsurface tension the problem exhibits small divisors, and is more difficult.