Mathematical Sciences: Geometric Stability Theory

9400894赫鲁晓夫斯基，这位研究者想要研究弗罗贝尼乌斯自同构的模型理论。这项工作已经开始，在两条战线上。一方面，他提出了一个类似于Lang-Weil估计的猜想，但更一般，这将决定Frobenius的一阶理论。另一方面，他正在与几位同事一起，沿着谢拉和齐尔伯的思路，对所讨论的差分场理论进行基本模式理论分析。综上所述，这些应该会给这个问题带来相当大的启发；将会有应用于有限单群，也许还会应用于一般有限群。上段所述的工作是数理逻辑领域中被称为模型理论的大趋势的一部分。几十年来，这门学科一直在自主发展。由于没有外部压力，它可以提炼出对代数结构的不同观点，并达到一定程度的技术成熟度。然而，这是在实验室条件下，在实际应用中过于严格的假设(“稳定性”)下完成的。用更复杂的结构--尤其是数论和几何结构--来检验这一时期形成的思想的时机似乎已经成熟。这需要确定适合模型理论分析的结构，并执行为模型理论工具做准备所需的代数几何分析。它还要求将模型理论思想概括为超出构思它们的“稳定”背景。这些方法已经取得了一些成功，包括对具有高度对称性的有限结构的研究，以及对数论中某些猜想的证明。人们希望这只是这条路的开始。***

9400894 Hrushovski The investigator would like to study the model theory of the Frobenius automorphism. This work has already started, on two fronts. On the one hand, he formulated a conjecture analogous to the Lang-Weil estimates, but more general, that would determine the first order theory of the Frobenius. On the other hand, together with several coworkers, he is in the process of carrying out the fundamental model theoretic analysis of the theory of difference fields in question, along the lines of Shelah and Zilber. Put together, these should throw considerable light on the issue; there will be applications to finite simple groups and perhaps to finite groups in general. The work described in the above paragraph forms part of a general trend in the area of mathematical logic known as model theory. For several decades, this subject developed autonomously. The lack of external pressure allowed it to refine distinct points of view on algebraic structures, and to reach a degree of technical maturity. However, this was done as it were in laboratory conditions, under assumptions ("stability") that are too restrictive in practical applications. The time seems ripe to test the ideas developed in this period against more complex structures, in particular, number theoretic and geometric ones. This requires identifying structures appropriate for model theoretic analysis and performing an algebro-geometric analysis needed to prepare for the model theoretic tools. It also requires generalizing the model theoretic ideas beyond the "stable" context in which they were conceived. Some success has been had with these methods, including the investigation of finite structures with a high degree of symmetry, and the proof of certain conjectures in number theory. One hopes that this is only the beginning of the road. ***