Tame Geometry

研究的基本问题是确定半代数几何的哪些展开式可以(或应该)被认为是驯服的，并且可以用模型论技术来处理。这项研究的中心目标之一是更好地理解整数集的不可定义性对分数可定义性的几何后果。为此，我们将应用实数分析、几何测度论和模型论的方法，其中一些方法还没有被用于实域的扩展的研究。这一研究不仅对序域中驯服的基础研究具有重要意义，而且对控制论和实解析几何中的问题也具有重要意义。作为数理逻辑的一个分支，o极小几何的研究可以作为研究驯服几何对象的框架。在过去的二十年里，研究人员证明，实解析几何中的许多经典现象都属于o-极小的框架。这不仅导致了几何学的进步，也导致了其他数学分支的进步，如李理论和数论，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的进步。不幸的是，o-极小只能用来模拟至少局部有限的现象(更准确地说，局部只有有限多个连通分量)。许多自然几何对象，如螺旋线或分形图，不具有这种有限性，因此已知的技术不适用于这种对象。希罗尼的目标是通过研究驯服来克服这一局限性，即使在局部有限的设置之外也是如此。

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.