Mathematical Sciences: Geometrical Studies of Superstable Theories

比希勒教授在数理逻辑方面的研究是在模型论的一个分支——稳定性理论方面。近年来，利用稳定性层次理论系统地研究了一些一般性的模型理论问题。尽管问题本身可能不涉及稳定性理论概念，但单独考虑ω -稳定、超稳定等情况，就需要使用大量的技术机制。这就是沃特猜想的情况，这一猜想被希拉在-稳定理论中证明了。下一步是考虑所有的超稳定理论，但事实证明，对于在-稳定情况下使用的方法来说，这太困难了。所需要的是一种分析，这种分析对其中的方式是敏感的，例如，将秩为α的集合拼凑在一起形成秩为α + 1的集合。比希勒将继续用这种几何方法来解决这个问题，这种方法已经产生了显著的结果。

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.