Analytic and smooth functions definable in o-minimal structures

这个项目是关于O-最小结构的，它使用模型论（逻辑的一部分）的思想来隔离欧几里得空间子集的驯服类和这些集合之间的良好行为函数。该项目的重点是更好地理解这些功能的行为，以及它们是如何从更基本的功能中建立起来的。最初的工作将涉及使用逻辑技术来证明，在许多情况下，从逻辑的角度来看，所讨论的结构是简单的。这将与解析几何思想结合起来，从奇点的解决方案中获得一个简单的解析描述这些结构中的功能。进一步的研究将使用强大的工具，从解析几何尚未被用于在这个模型的理论设置。该项目正在进行的一个重要方面是寻找在分析或数论中自然产生的函数的例子，理论可以应用于这些例子。此外，该理论将被应用于围绕决定的问题，算法，如果formalstatements是真的，在某些结构。

This project is about o-minimal structures, which use ideas from modeltheory (a part of logic) to isolate tame classes of subsets ofEuclidean spaces and well behaved functions between these sets. Thefocus of the project is on a better understanding of the behaviour ofthese functions, and how they are built up from more basic functions.Initially the work will involve using techniques from logic to provethat, in many cases, the structures in question are simple, from alogical point of view. This will then be combined with analyticgeometric ideas from resolution of singularities to obtain a simpleanalytic description of the functions in these structures. Thenfurther study will be made using powerful tools from analytic geometrywhich have yet to be used in this model theoretic setting. An important ongoing aspect of the project is to find examples offunctions which arise naturally in analysis or number theory, to whichthe theory can be applied. In addition, the theory will be applied tological problems around deciding, algorithmically, if formalstatements are true in certain structures.