The rigidity question was first studied for the automorphisms of Boolean algebras. A groundbreaking result of Shelah shows that it is consistent with the usual axioms of mathematics (ZFC) that all automorphisms of the Boolean algebra P(N)/Fin are trivial, in the sense that they are implemented by almost permutations of the natural numbers N. While assuming the Continuum Hypothesis, a transfinite induction due to Rudin shows that there are many nontrivial automorphisms of P(N)/Fin. Using the stone duality, the results of this kind can be translated in the category of (zero-dimensional locally compact and Hausdorff) topological spaces. Motivated by a question of Brown-Douglas-Fillmore, the rigidity question was studied in the non-commutative settings for the corona of C*-algebras. It is proved by Philips-Weaver that assuming the Continuum Hypothesis the Calkin algebra has outer automorphisms. On the other hand Farah showed that the Open Coloring Axiom implies that all the automorphisms of the Calkin algebras are inner (implemented by almost unitary elements of B(H)). In my talks we will take a closer look at Farah's result and study the rigidity question for the isomophisms between the corona of Finite-Dimensional Decomposition algebras (reduced products of matrices) and its relevance to Farah's result. We will also look at the later question in more general setting, motivated by the more recent results of Farah-Shelah about the Boolean algebra counterparts.

Saeed obtained his Ph.D. this spring at York University in Toronto under the guidance of Ilijas Farah and now is a new posdoc at IM PAN. We plan a series of his three talks under the above title.