Mathematical Sciences: Geometrical Stability Theory

史蒂文·A·比克勒的研究项目是稳定性理论，这是模型理论的一个子领域。近年来，人们通过稳定性层次系统地研究了几个一般性的模型理论问题。即使问题本身可能不涉及稳定性理论的概念，但单独考虑欧米伽稳定、超稳定等情况，也会发挥大量的技术机械。这就是沃特猜想的情况，这一猜想被谢拉用欧米伽稳定理论证明了。下一步是考虑所有的超稳定理论，但事实证明这太难了。比克勒将根据超稳定理论的“几何”性质来解决这个问题。第一个要考虑的情况是，当宇宙的谢拉等级为1的时候。比切勒已经在这个问题上取得了进展。

Steven A. Buechler's research project is in stability theory, a subfield of model theory. In recent years several general model- theoretic problems have been systematically approached via the stability hierarchy. Even though the problem itself may not involve stability-theoretic notions, considering separately the cases: omega-stable, superstable, etc., brings into play a mass of technical machinery. Such is the case with Vaught's conjecture, which was proved by Shelah for omega-stable theories. The next step is to consider all superstable theories, but this has proven to be too difficult. Buechler will approach this problem by separating the superstable theories according to their "geometrical" properties. The first case to consider is when the universe has Shelah rank 1. Buechler has already made progress on this problem.