Lange will explore several subjects that require solving problems in pure computability theory, some that are purely model-theoretic, and some that are a mix of the two. For example, Lange is studying the structure consisting of the computably enumerable (c.e.) sets under inclusion, an object of intense study in classical computability theory. The problems she plans to study focus on when a given c.e. set is mapped by an automorphism of this structure to a set that computes the halting problem. On the other hand, Lange is examining the computational complexity of structures associated with real closed fields, generalizations of the ordered field of the reals. Many of these questions cannot be answered without first solving purely algebraic or model theoretic problems.
The aim of computable structure theory, Lange's area of research, is to examine mathematical structures from an algorithmic perspective. There are many problems in mathematics for which there is no way to compute a solution. Knowing a problem is not computably solvable, however, provides only limited information about its algorithmic complexity. Given two problems, if one can compute a solution to the first problem from a solution to the second, the second problem is at least as computationally difficult as the first. Lange and others in the field work to precisely calibrate the algorithmic difficulty of a mathematical problem by finding its place in the standard computational framework that grew out of Turing's work. A critical aspect of computable structure theory is understanding the relationship between definability and computability. Definability refers to the ability to describe aspects of a mathematical object using a formal language. Thus, answering questions in computable structure theory often furthers the understanding of the mathematical objects under consideration.
