O-minimality is a branch of model theory dealing with "geometrically tame" structures. In 2006 Pila and Wilkie discovered a deep "counting theorem" about the asymptotic density of rational points in sets definable in an o-minimal structure. Since then, a deep link has been evolving between o-minimality and various aspects of Diophantine geometry, transcendence theory, Hodge theory etc. The 2006 paper by Pila and Wilkie also contained a conjectural sharpening of the counting theorem for certain structures. In the past few years there has been a lot of progress around this conjecture, and such sharper forms have recently played a major role in the proof of the Andre-Oort conjecture for general Shimura varieties. In this talk I'll try to give an overview of this subject without assuming familiarity with o-minimal geometry.

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