Analytic and smooth functions definable in o-minimal structures

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.