Mathematical Sciences: Geometric Stability Theory

9400894 Hrushovski 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。 一方面，他提出了一个类似于朗-韦尔估计的猜想，但更普遍，这将决定弗罗贝尼乌斯的一阶理论。 另一方面，他正在与几位同事一起，沿着 Shelah 和 Zilber 的思路，对所讨论的差分场理论进行基本模型理论分析。 总而言之，这些应该可以很好地阐明这个问题。将会应用于有限单群，也许还有一般的有限群。 上一段中描述的工作构成了数理逻辑领域称为模型理论的总体趋势的一部分。 几十年来，这个学科自主发展。 由于缺乏外部压力，它能够提炼出关于代数结构的独特观点，并达到一定程度的技术成熟度。 然而，这是在实验室条件下完成的，假设（“稳定性”）在实际应用中限制太多。 时机似乎已经成熟，可以用更复杂的结构（特别是数论和几何结构）来测试这一时期发展的思想。 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好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其构思的“稳定”背景之外。 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. ***