Model theory of valued fields as examples of theories with NIP

有价值领域的模型理论作为 NIP 理论的例子

• 批准号: 238875-2011
• 负责人: Haskell, Deirdre
• 金额: $ 0.8万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570511
• 关键词: Model; theory; valued; fields; examples

Much of the power (and much of the difficulty) of mathematics comes from its abstractness. A mathematician will observe many different situations - the shape of petals in a flower, the diffusion of particles through a membrane, the flow of water in the confluence of rivers - and look for the features they have in common. By studying these features in an abstract way, we can understand what properties the different situations have which arise simply because they are examples of the abstract setting. By contrast, we can also understand what properties are special to each example. In model theory, this process of abstraction is taken one step further. We study different mathematical objects by looking at their axiomatic properties, and thus try to determine features of those objects which come from their common abstract characteristics. The particular mathematical objects that are the subject of this research proposal are fields with a valuation. A valuation is a coarse measure of size. It might measure the amount of divisibility of a number by a given prime, or the behaviour of a function near a singularity, or a size of infinity. The goals of the research described in this proposal are three-fold. The first is to work directly with fields with a valuation. Specific questions include quantifier elimination and elimination of imaginaries, which can both be understood as questions about how complicated are the sets which can be defined. The second is to study more general theories, called NIP theories, which have features in common with valued fields. The third is to strive to make this work more accessible to graduate students by writing a textbook.

数学的大部分力量（和大部分困难）来自于它的抽象性。一个数学家会观察许多不同的情况--花瓣的形状，粒子通过膜的扩散，河流汇合处的水流--并寻找它们的共同特征。通过以抽象的方式研究这些特征，我们可以理解仅仅因为它们是抽象背景的例子而出现的不同情况具有什么性质。通过对比，我们也可以理解每个例子的特殊属性。 在模型论中，这个抽象过程更进一步。我们研究不同的数学对象，通过看他们的公理性质，从而试图确定这些对象的特征，来自他们的共同的抽象特征。作为本研究建议主题的特定数学对象是具有估值的领域。估值是对规模的粗略衡量。它可以测量一个数被给定素数整除的程度，或者函数在奇点附近的行为，或者无穷大的大小。 本提案中所述的研究目标有三个方面。第一种是直接与具有估值的字段合作。具体问题包括量词消去和量词消去，这两个问题都可以理解为关于可以定义的集合有多复杂的问题。第二是研究更一般的理论，称为NIP理论，它与有价值的领域有共同的特征。三是通过编写教材，努力使这项工作更容易为研究生所用。