Mathematical Sciences: Superstability in Model Theory

点击翻译按钮获取中文摘要

Abstract DMS 9626112 Ambar Chowdhury, U. Connecticut Classification theory is concerned with determining whether a given class of mathematical objects admits a "structure theory". In the best cases, a structure theory assigns certain dimensions which determine each object in the class up to isomorphism. The class of vector spaces over a given field, with the usual notion of dimension, is a simple example of a good case. More generally a structure theory aims to describe how each member of the class is built from smaller objects in the same class, or even from simpler objects belonging to a second, well understood class. Shelah has isolated certain properties of a countable, complete first order theory which are necessary and sufficient for the class models of T to admit a structure theory. One key property is that of being superstable, a guarantee that certain subsets of the models of T can be viewed as pregeometries, and thus carry the desired dimensions. Investigating the precise nature of the aforementioned pregeometries, and the way in which they interact, can yield very deep information about the fine structure of the models; such investigation has become known as geometric stability theory. Chowdhuiy will apply the methods of geometric stability theory to gain further information about superstable theories. In particular, his work is aimed at extending Shelah's classification results to uncountable theories. This should result in new structural information regarding the models of superstable theories, countable or not. In addition, Chowdhury plans to examine which properties of superstable theories can be generalised to a broader class, called supersimple theories. Supersimple theories, though not necessarily stable, exhibit some of the important characteristics of superstable theories. In model theory one studies those mathematical objects (called 'models") which satisfy a given collection of axioms. While one may think of a group as the set of transformations , or symmetries, of a particular object - such as an organic molecule, the general class of objects known as groups are precisely the models of the three group axioms. In algebra, one attempts to understand complicated objects (e.g., groups) in terms of how they are built from well-understood objects; when this succeeds, this is called a "structure theory". In model theory, one considers a general set of axioms and the collection of models, and asks whether there are properties inherent in the set of axioms which guarantee the existence of a structure theory. Rather remarkably, it turns out that certain abstract properties of a set of axioms do indeed imply certain structural features for the corresponding objects; one such property is called "superstability". The principal investigator proposes to continue the study of superstability (and its variations) and of its structural implications for a broader array of objects, one which includes interesting and important mathematical classes.