Tame Geometry

调查的基本问题是，以确定哪些扩展的半代数几何可以（或应该）被视为驯服，可以处理的模型理论技术。这项调查的中心目标之一是实现更好地理解的几何后果的不可定义的一组整数上的可定义的分形。为了达到这个目标，我们将应用从真实的分析，几何测量理论和模型理论，其中一些尚未被用于研究的真实的领域的扩展的方法。本文的研究成果不仅对有序域中的驯服性的基础研究有重要意义，而且对控制论和实解析几何中的一些问题也有一定的指导意义。作为数理逻辑的一个分支，o-极小几何的研究可以作为驯服几何对象的研究框架。在过去的二十年里，研究者证明了许多实解析几何中的经典现象都属于o-极小的范畴。这不仅导致了几何学的进步，而且还导致了其他数学分支的进步，如李学和数论，以及经济学中的一般均衡理论和计算机科学中的神经网络等不同领域。不幸的是，o-极小性只能用于模拟至少局部有限的现象（更准确地说，局部只有100个连通分量）。许多自然的几何对象，如螺旋或分形不具有这种有限性属性，因此已知的技术不适用于这样的对象。Hierkini的目标是通过研究甚至在局部有限性之外的驯服来克服这种局限性。

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.