Mathematical Sciences: Geometric Stability Theory

[400894]研究者想研究Frobenius自同构的模型理论。这项工作已经在两个方面开始了。一方面，他提出了一个类似于朗-韦尔估计的猜想，但更普遍，这将决定Frobenius的一阶理论。另一方面，他与几位同事正在沿着Shelah和Zilber的思路，对所讨论的差异场理论进行基本的模型理论分析。把这些放在一起，应该能让我们对这个问题有相当大的了解；将会应用于有限单群或许也会应用于一般的有限群。上述段落中描述的工作构成了数学逻辑领域中被称为模型理论的一般趋势的一部分。几十年来，这门学科自主发展。由于缺乏外部压力，它得以完善关于代数结构的不同观点，并达到一定程度的技术成熟。然而，这是在实验室条件下完成的，假设（“稳定性”）在实际应用中限制太大。检验这一时期发展起来的思想在更复杂的结构上的时机似乎已经成熟，特别是在数论和几何结构上。这需要识别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。它还需要将模型理论思想推广到它们被构想的“稳定”环境之外。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。人们希望这只是道路的开始。＊＊＊

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. ***