Mathematical Sciences: General Frameworks for Classification Theory

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Professor Buechler's proposed research is in stability-theoretic classification theory, a subfield of model theory. In recent years, methods developed in the context of stability theory have been used to analyze structures such as pseudo-finite fields, pseudo-algebraically closed fields, difference fields, and quadratic forms over finite fields. This research has yielded applications to arithmetic number theory (due largely to Hrushovski). Model-theorists now have a good understanding of how these dependence relations fit in a general framework. Buechler proposes to generalize techniques from the geometrical stability theory of superstable theories to this broader class. This research is likely to give insight into the model-theoretic properties of bilinear forms and groups definable in structures such as those mentioned above. Buechler also intends to set a more general framework for arguments based on ranks and dependence relations. Much of mathematics involves the classification of an abstract collection of objects in terms of concrete, well-understood objects. The classification of finite simple groups is one instance of this principle. Model theorists study classification results in an abstract setting, addressing questions such as: When can we prove that a classification theorem exists, even when the particular theorem is unknown? How can we tell when a particular classification theorem is the best possible result for a given collection of objects? When are two classification theorems special cases of a more general result? In the past twenty years model theorists have gone a long way towards answering these questions for collections of objects axiomatized in a relatively simple manner. While this research is carried out in abstract setting, it has spawned concrete results in differential algebra and arithmetic number theory. Buechler proposes to expand the mathematical context in which these results apply. He will also find a more abstract notion of classification theorem which encompasses several existing examples. This may lead to a greater understanding of results related to Fermat's Last Theorem.