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).
