The basic question of the investigation is to determine which expansions of semialgebraic geometry can (or should) be considered as tame and can be handled by model theoretic techniques. One of the central goals of this investigation is to achieve a better understanding of the geometric consequences of the non-definability of the set of integers on the definability of fractals. Towards this goal, we will apply methods from real analysis, geometric measure theory and model theory, some of which have not yet been used in the study of expansions of the real field. The insights of this research are not only important in the fundamental study of tameness in ordered fields, but should also prove useful to questions in control theory and real-analytic geometry.
The study of o-minimal geometry, a branch of mathematical logic, can be considered as a framework for studying tame geometric objects. In the last two decades researchers proved that many classical phenomena from real-analytic geometry fall into the framework of o-minimality. This has led to advances not only in geometry, but also in other branches of mathematics like Lie theory and number theory, and in such diverse fields as general equilibrium theory in economics and neural networks in computer science. Unfortunately, o-minimality can only be used to model phenomena that are at least locally finite (more precisely, locally having only finitely many connected components). Many natural geometric objects like spirals or fractals do not have this finiteness property and hence the known techniques are unapplicable to such an object. Hieronymi aims to overcome this limitation by studying tameness even outside the setting of local finiteness.
