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，其中包括可能存在一个超稳定的非微分闭微分场；2)一些涉及可数和不可数稳定理论的模型数目的问题；3)在充分丰富的o-极小结构中定义场的可能性，以及可在o-极小结构中定义的群的识别。数理逻辑中的模型理论的子领域涉及公理系统，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来回答，而不真正考虑它们的内容。有时可以通过这种方式学到很多东西，这是令人惊讶的，因为它往往似乎是不劳而获。模型理论家们已经发展出一门丰富的学科，甚至能够阐明诸如群论和场论等代数中研究得很好的主题。

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.