In this talk I will discuss advances in birational geometry in recent years. Much of the relevant results are concerned with boundedness properties of linear systems and of singularities which are in turn closely related with moduli theory of algebraic varieties

A feature of solutions of a (generally nonlinear) field theory can be called "universal" if it is independent of side conditions like initial data. I will explain this phenomenon in some detail and then illustrate it in the context of the sine-Gordon equation, a fundamental relativistic nonlinear wave equation. In particular I will describe some results (joint work with R. Buckingham) concerning a universal wave pattern that appears for all initial data that crosses the separatrix in the phase portrait of the simple pendulum.