Mathematical Sciences: Geometrical Studies of Superstable Theories

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Professor Buechler's research in mathematical logic is in stability theory, a subfield of model theory. In recent years several general model-theoretic problems have been systematically approached via the stability hierarchy. Even though the problem itself may not involve stability-theoretic notions, considering separately the cases: omega-stable, superstable, etc., brings into play a mass of technical machinery. Such is the case for Vaught's conjecture, which was proved by Shelah for omega-stable theories. The next step is to consider all superstable theories, but this has proved too difficult for the methods used in the omega-stable case. What is required is an analysis which is sensitive to the way in which, e.g., sets of rank alpha are pieced together to form sets of rank alpha plus one. Buechler will continue with this geometrical approach to the problem, which has already yielded significant results.