José G. Llorente (Universidad Complutense de Madrid)
Variations on Picard
Any holomorphic function in the complex plane whose range omits two complex numbers is constant. This is Picard’s little theorem (1879), one of the most striking results in classical Complex Analysis. Since Picard’s original proof, a great variety of approaches, reinterpretations and fascinating links with other areas have contributed to enrich the scope of Geometric Function Theory.
In this mostly expository talk we will briefly explore four variations on Picard’s theorem, related to i) normal families of holomorphic functions, ii) Fermat’s functional equation, iii) the Gauss map of minimal surfaces and iv) the real formulation of Picard’s theorem, including a recent extensiоn in terms of the range of planar harmonic maps.