Mathematical Sciences: Topics in Model Theory

Pillay打算研究：1)微分场模型理论中的一些问题，其中包括超稳定非微分闭微分场存在的可能性；2)涉及可数稳定理论和不可数稳定理论模型数目的若干问题；3)在足够丰富的0 -极小结构中定义域的可能性，以及0 -极小结构中可定义群的识别。数学逻辑中的模型论子领域关注公理系统，特别是关于建立在公理系统上的理论的问题，这些问题可以纯粹通过考虑公理的形式而不真正考虑其内容来回答。令人惊讶的是，有时通过这种方式可以学到很多东西，因为它往往似乎是不劳而获。模型理论家已经发展了一个丰富的主题，并且能够阐明代数中诸如群理论和场理论等研究得很好的主题。

Pillay intends to investigate: 1) some problems in the model theory of differential fields, among them the possible existence of a superstable non-differentially closed differential field; 2) some problems involving the number of models of both countable and uncountable stable theories; 3) the possibility of defining a field in sufficiently rich o-minimal structures, and the identification of groups definable in o-minimal structures. The subarea of model theory within mathematical logic concerns itself with axiomatic systems, particularly questions about theories founded on systems of axioms, which can be answered purely from considering the forms of the axioms without real regard for their content. It is surprising how much can sometimes be learned in this way, for it often seems to be getting something for nothing. Model theorists have developed a rich subject and have been able to shed light even on such well-studied topics in algebra as the theory of groups and the theory of fields.