Undecidable Theories and Global Properties of Structures

Nies打算研究范围广泛的基本理论和结构，源于可计算理论，复杂性理论，代数和可计算分析。自从哥德尔和塔斯基的工作以来，一个给定的基本理论是否可决定的问题一直是逻辑学家的中心兴趣。Nies将在非abel自由群理论和在Solovay支配可约性下比较递归可数实数的度结构方面研究这个问题。作为主要工具，他将使用一阶公式编码。这种方法不仅可以得到结构的基本理论的不确定性，而且还可以得到结构本身（而不是其理论）的进一步全局性质。该方法得到的典型结果是可定义性结果和自同构的局限性。Nies将研究这些性质，特别是递归可枚举图灵度的中心结构。Nies的工作通过分析来自可计算性理论和分析等不同领域的共同方法，将统一的方面引入了现代数学的广阔而分散的领域。这些方法被称为编码方法。它们包括用一阶公式在第二类对象（正在研究的结构）中表示一类对象。一阶语言是数学逻辑和计算机科学的核心。从某种意义上说，它们是自然语言片段的形式化。结构的一阶理论是关于该结构的所有事实的集合，这些事实可以用该语言表达。例如，无穷多个素数的存在是自然数结构（有加法和乘法）的一阶事实。在某种程度上，尼斯研究了一阶理论是否可决定的问题。此外，他使用编码方法来探索结构的全局属性，比如对称性的存在。在一些情况下，研究的结构被认为是问题区域的中心（如代数中的自由群或可计算理论中的递归可枚举图灵度结构）。

Nies intends to study a wide range of elementary theories and structures stemming from computability theory, complexity theory, algebra, and computable analysis. Ever since the work of Godel and Tarski, the question of whether a given elementary theory is decidable has been of central interest to logicians. Nies will study this question for the theories of nonabelian free groups and for the degree structure of recursively enumerable reals compared under Solovay's domination reducibility. As a main tool, he will use coding with first-order formulas. This method yields not only undecidability of the elementary theory of a structure, but also further global properties of the structure itself (rather than its theory). Typical results obtained by this method are definability results and limitations on automorphisms. Nies will study such properties, in particular for the central structure of recursively enumerable Turing degrees. Nies' work introduces a unifying aspect into the vast and diverging area of modern mathematics, by analyzing through common methods structures from such different domains as computability theory and analysis. These methods are called coding methods. They consist of representing an object of one kind in an object of a second kind (the structure under investigation) using first order formulas. First-order languages are central to mathematical logic and computer science. In a sense, they are a formalization of a fragment of natural language. The first-order theory of a structure is the collection of all facts about the structure which can be expressed in that language. For instance, the existence of infinitely many prime numbers is a first-order fact about the structure of natural numbers (with addition and multiplication). In part, Nies investigates the question whether first-order theories are decidable. Moreover, he uses coding methods to explore global properties of the structures, like the existence of symmetries. In several cases, the structures investigate d are considered central for the area in question (like free groups in algebra or the structure of recursively enumerable Turing degrees in computability theory).