Tomasz Szemberg: New developments on Waldschmidt constants

In recent years there has been increasing interest in asymptotic methods in geometry and algebra. Waldschmidt constants were introduced in context of diophantine approximation about 60 years ago by Waldschmidt and Chudnovsky, considered by Demailly at the beginning of the 80's in the context of complex analysis and rediscovered recently by Dumnicki and Harbourne in the realms of commutative algebra. I will present the algebraic face of Waldschmidt constants, recall their basic properties (which I reported on last November) and focus on recent results on lower bounds on these constants obtained jointly with Dumnicki and Szpond for sets of very general points in projective spaces and on related results for special configurations of points in the projective plane obtained also recently by the Cracow Group.