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Resonance is one of the most versatile concepts of science. It is present under various aspects in astronomy, physics, technology, musics, and, of course, mathematics. At resonance, the response of a vibrating system to a periodic excitation becomes unbounded; resonance can destroy the stability of a bridge or of the solar system. The mathematical study of resonance is closely linked to the concept of spectrum, a name coined in optics by Newton in 1672 and part of the mathematical language since the end of the XIXth century only. The presence of nonlinear terms in an equation modifies the resonance phenomenon in essential ways, and the concept of spectrum can be extended to some classes of nonlinear operators, leading to new resonance conditions. The lecture describes some recent results relating spectra and nonlinearities in differential equations.
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