Mathematical Sciences: The Fine Structure of Superstable Theories

数学科学：超稳定理论的精细结构

• 批准号: 9223767
• 负责人: Steven Buechler
• 金额: $ 9.15万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9223767&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Fine; Structure; Superstable

Professor Buechler's research is in stability theory, a subfield of model theory. In recent years several general model-theoretic problems have been systematically approached via the stability-theoretic hierarchy. Even though a problem itself may not involve stability-theoretic notions, considering separately the cases: totally transcendental, superstable, etc., brings into play a mass of technical machinery. Such is the case for Vaught's conjecture, which was proved by Shelah for totally transcendental theories and (over the past 5 years) by Buechler for the superstable theories of finite rank. Buechler intends to continue using the methods of geometrical stability theory to gain insight into Vaught's conjecture for all superstable theories. The geometrical methods bring modules into the picture, allowing for the use of representation theory and other algebra to pin down further the models of these theories. This research lies in the general setting of abstract "Classification Theory." In many diverse branches of mathematics researchers attempt to code each element of a class of structures with an element of a "simpler" class of structures. The original class is then more easily analyzed, using the more detailed information that one can obtain for this simpler class. Classification Theory is the abstract study of such codings. Researchers have shown that it is possible to determine when such a simplification exists by using only rather basic information about the original class. Buechler intends to study when an abstract class can be simplified down to a very precise class of objects called "modules," which have been studied in great detail by algebraists.

Buechler教授的研究是稳定理论，这是模型理论的一个子领域。 近年来，通过稳定理论层次结构系统地解决了一些常规模型理论问题。 即使问题本身可能不涉及稳定理论观念，但要分别考虑这些案例：完全先验，超级明星等，也可以发挥大量的技术机械。 Vaught的猜想就是这种情况，Shelah证明了完全先验理论，并且（在过去的5年中）Buechler为有限的级别的超级巨星理论所证明。 Buechler打算继续使用几何稳定理论方法，以了解Vaught对所有超级巨星理论的猜想。 几何方法将模块带入图片中，从而使使用代表理论和其他代数进一步固定这些理论的模型。 这项研究在于抽象“分类理论”的一般环境。 在许多不同的数学分支中，研究人员试图编写具有“简单”类别类别的元素的一类结构元素。 然后，使用更简单的类可以获得的更详细的信息，可以更轻松地分析原始类。 分类理论是对此类编码的抽象研究。 研究人员已经表明，仅通过仅使用有关原始类的相当基本信息来确定这种简化何时存在。 Buechler打算研究何时可以将抽象类简化为一个非常精确的对象，称为“模块”，这些对象已由代数主义者进行了详细的详细研究。