Mathematical Sciences: The Fine Structure of Superstable Theories

点击翻译按钮获取中文摘要

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.