My encounters with Alex Müller and the perovskites

This paper is dedicated to the memory of Professor K. Alex Müller. After describing our personal and scientific encounters since 1974, I concentrate on the many puzzles which appeared in our discussions and collaborations, involving the interplay between theory and experiments on the critical behavior of cubic perovskites which undergo (second or first) order transitions to a lower symmetry phases (trigonal or tetragonal). The conclusion, reached only very recently, is that (although beginning with the same cubic symmetry) the two types of transitions belong to two distinct universality classes: under [100] stress, the cubic to trigonal transition exhibits a tetracritical phase diagram, with ‘cubic’ exponents, while the cubic to tetragonal transition exhibit an ‘intermediate’ bicritical phase diagram, but asymptotically the bicritical point turns into a triple point, with three first order lines. To test these conclusions, it is suggested to measure the effective critical exponents as the temperature approaches criticality.