Model theory of valued fields as examples of theories with NIP

数学的大部分力量(和许多困难)来自于它的抽象性。数学家将观察许多不同的情况--花瓣的形状，微粒在膜中的扩散，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，我们可以理解不同的情况具有什么性质，仅仅因为它们是抽象环境的例子而出现。相比之下，我们还可以了解每个示例的特殊属性。 在模型理论中，这一抽象过程更进一步。我们通过观察不同的数学对象的公理化性质来研究不同的数学对象，从而试图从它们共同的抽象特征中确定这些对象的特征。作为这项研究建议的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。它可以衡量一个数被给定素数整除的程度，或者一个函数在奇点附近的行为，或者无穷大的大小。 这项提案中描述的研究目标有三个方面。第一种方法是直接处理有估值的油田。具体问题包括量词剔除和想象剔除，这两个问题都可以理解为可以定义的集合有多复杂的问题。第二是学习更一般的理论，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，努力让这项工作更容易为研究生所接受。

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.