Strongly Minimal Sets and Geometry
Strongly Minimal Sets and Geometry
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极小集和几何
DOI：
10.1007/978-3-662-22108-2_12
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
D. Marker
中科院分区：
文献类型：
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作者：
D. Marker
这些讲座将简要介绍强极小集的模型理论。前两节将介绍强极小集的基本组合几何。在第三节和第四节中，我们将展示强极小集的准几何如何检测环境代数结构的存在。最后，我们将在Hrushovski对函数域的Mordell-Lang猜想的证明([HI])中展示这些想法是如何结合在一起的。在几何模型理论中，强极小集仅仅是故事的开始。我希望通过专注于强极小集，我可以对这个重要而美丽的主题给出一个相当自成体系的介绍。我们只假定读者熟悉[CK]或[S]中给出的ω稳定理论的处理。几何模型理论在[PI]中得到了充分的发展，我自己对几何模型理论的欣赏来得很慢。我非常感谢阿南德·皮莱、伊丽莎白·布斯卡伦和约翰·鲍德温，他们用无数次的对话启发了我。
These lectures will provide a brief introduction to the model theory of strongly minimal sets. The first two sections will develop the basic combinatorial geometry of strongly minimal sets. In sections three and four we will show how the pregeometry of the strongly minimal set detects the presence of ambient algebraic structure. Finally we will show how these ideas come together in Hrushovski's proof ([HI]) of the Mordell-Lang conjecture for function fields. Strongly minimal sets are just the beginning of the story in geometric model theory. My hope is that by concentrating on strongly minimal sets I can give a reasonably self-contained introduction to this important and beautiful subject. We assume only that the reader is familiar with the treatment of ω-stable theories given in [CK] or [S]. A full development of geometric model theory is given in [Pi] My own appreciation of geometric model theory was slow in coming. I owe a great deal to Anand Pillay, Elisabeth Bouscaren and John Baldwin for the numerous conversations it took to enlighten me.