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Feynman amplitudes in perturbation theory form the basis for most predictionsin particle collider experiments. The mathematical quantities which occur asamplitudes include values of the Riemann zeta function and relate tofundamental objects in number theory and algebraic geometry. This talk reviewssome of the recent developments in this field, and explains how new ideas fromalgebraic geometry have led to much progress in our understanding ofamplitudes. In particular, the idea that certain transcendental numbers, suchas $\pi$, can be viewed as a representation of a group, provides a powerfulframework to study amplitudes which reveals many hidden structures.

Periods and Feynman amplitudes