Mathematical Sciences: Model Theory for Analytic Structures

研究者将继续研究与实数和复数相关的结构模型理论。特别是，他将专注于具有额外分析结构的展开中的可定义性问题。他过去五年的大部分工作都集中在这一领域。他将尝试证明以下性格的进一步结果。(1)Marker，Macintyre和van den Dries证明了具有指数和受限解析函数的实数的结构是O-极小的，并且允许去掉量词。O-极小性意味着可定义集有丰富的结构理论。(2)Marker和Pillay证明了含有加法的代数闭域的任何非平凡约化都足以恢复整个域的结构。(3)Marker已经证明，如果将任何非平凡实数结构加到复数上，则实数是可定义的。后两个结果可以被认为是对代数几何边界的一种尝试。第一种观点认为，弱化代数几何只会导致线性几何。第二种观点认为，推广代数几何会导致实代数几何的完全复杂化。虽然“可定义性”是一个逻辑概念，但事实往往证明，可定义性集合是自然产生的、非常重要的集合。模型论方法为它们的研究提供了新的工具。例如，在研究解析几何、动力系统、统计学和控制论时，可以在实数的结构中定义的具有指数和受限解析函数的集合自然出现。

The investigator will continue his research on the model theory of structures related to real and complex numbers. In particular, he will focus on definability questions in expansions with extra analytic structure. Most of his work over the past five years has been concentrated in this area. He will attempt to prove further results of the following character. (1) Marker, Macintyre, and van den Dries have shown that the structure of the real numbers with exponentiation and restricted analytic functions is O-minimal and admits elimination of quantifiers. O-minimality implies that there is a rich structure theory for the definable sets. (2) Marker and Pillay have shown that any non-trivial reduct of an algebraically closed field that contains addition is strong enough to recover the full field structure. (3) Marker has shown that if any non-trivial real structure is added to the complex numbers, then the real numbers are definable. These latter two results can be thought of as attempts to examine the boundary of algebraic geometry. The first says that weakening algebraic geometry leads to little more than linear geometry. The second says that extending algebraic geometry leads to the full complications of real algebraic geometry. While "definability" is a logical notion, often it turns out that the definable sets are ones which arise naturally and are of great importance. Model theoretic methods provide a new tool for their study. For example, the sets definable in the structure of the real numbers with exponentiation and restricted analytic functions arise naturally in studying analytic geometry, dynamical systems, statistics, and control theory.