Motives and Periods

Let X be a smooth algebraic variety defined over $Q$. Integrating algebraic differential forms over cycles on the complex variety associated with $X$ yields a class of complex numbers called ‘periods’. Grothendieck's period conjecture predicts what the set of all algebraic relations between these periods should be. I will explain this conjecture and discuss the scant evidence we have for it. Depending on time and mood of the audience, I will try to explain an extension of Grothendieck's period conjecture to ‘exponential periods’ (joint work with J. Fresán) and its connection with the Siegel-Shidlovskii theorem.