Principe d'Heisenberg et fonctions positives
  Starting from a problem in number theory, the article investigates the properties of the couples of Fourier transforms on the real line, f and g, real and even, f >= 0 out of an interval (-a, a) and f(0) < 0, g >= 0 out of an interval (-b, b) and g(0) < 0 . How small the product ab can be ? There is a strictly positive lower bound, the exact value is not known. The same problem is considered in several dimensions (where it is related to number theory, as the article points out). 
Jean Bourgain (IAS), Laurent Clozel (LM-Orsay), Jean-Pierre Kahane   (LM-Orsay)