RUI: Research in O-minimality and Related Topics

该项目处理有关0 -极小性、0 -极小性的扩展和有限结构类的问题。一些与0 -极小性有关的问题涉及到原型0 -极小结构的扩展，以及其定义域为实数阶型的结构。另一些则关注可在0 -极小结构中定义的阿贝尔群，或开发微分和代数拓扑方法和工具的0 -极小类似物。关于o-极小性扩展的问题特别涉及弱o-极小性、局部o-极小性，以及与Morley秩类似的有限秩有序结构的模型理论的发展。该项目的第三个主要课题涉及具有尺寸和测量的有限结构类。这项工作的目的是发展有限结构类的模型理论，与无限结构的主流模型理论类似。迄今为止取得的结果和已发现的例子表明，有许多工作要做。上述研究涉及数学逻辑的主要子领域之一模型论。模型理论家研究可以用形式数学语言（如谓词逻辑）表示的熟悉的数学结构的性质。这种独特的观点可以提供对这种结构的见解和理解，否则可能会被证明是难以捉摸的。这个项目的一个方面侧重于结构，包括和表现的重要方式，如实数的有序域，也就是说，实数与多项式和代数函数一起在一年级微积分中学习，并描述许多现象。模型理论在过去十年中取得的许多重大进展中发挥了关键作用。这些加深了我们对数学科学不同领域中熟悉的数学系统的理解，如实函数的分析和几何、神经网络和关系数据库理论。在经济学中也有应用。这个项目的第二个主要方面是处理有限结构类。一般来说，有限结构是计算机科学的核心：任何数据库都可以被解释为本文所研究的有限结构，而在密码学中，一类被称为有限域的有限结构尤为重要。

The project deals with questions concerning o-minimality, extensions ofo-minimality, and classes of finite structures. Some of the problems having to do with o-minimality relate to expansions of archetypal o-minimal structures and structures whose domain has as its order type that of the real numbers. Other have as their focus abelian groups definable in o-minimal structures or the development of o-minimal analogues of differential and algebraic topological methods and tools. Problems concerning extensions of o-minimality have to do in particular with weak o-minimality, local o-minimality, and, in analogy with Morley rank, the development of a model theory for ordered structures of finite rank. The third main topic of the project involves classes of finite structures with dimension and measure. This work has as its aim the development of a model theory for classes of finite structures that is in analogy with mainstream model theory for infinite structures. The results obtained to date and the examples that have been found suggest that there is much to be done.The research outlined above concerns model theory, one of the principal subfields of mathematical logic. Model theorists study properties of familiar mathematical structures that can be expressed in a formal mathematical language such as predicate logic. This distinctive point of view can provide insights and understanding into such structures that otherwise might prove elusive. One aspect of this project focuses on structures that include and behave in important ways like the ordered field of real numbers, that is, the real numbers together with the polynomial and algebraic functions that are studied in first-year calculus and describe many phenomena. Model theory has played a key role in many of the significant advances that have been made in the last ten years. These have deepened our understanding of familiar mathematical systems in such diverse areas of the mathematical sciences as the analysis and geometry of real functions, neural nets, and relational database theory. Applications also have been made in economics. A second principal aspect of the project deals with classes of finite structures. Finite structures in general are central to computer science: any database can be construed as a finite structure in the sense in which they are studied in here, and a particular class of finite structures called finite fields are especially important in cryptology.