Recent results for C^r-parameterizations and diophantine applications

Both in the real and in the p-adic case, I will talk about recent results about C^r-parameterizations and their diophantine applications.  In both cases, the dependence on r of the number of parameterizing C^r maps plays a role. In the non-archimedean case, we get as an application new bounds for rational points of bounded height lying on algebraic varieties defined over finite fields, sharpening the bounds by Sedunova, and making them uniform in the finite field. In the real case, some results from joint work with Pila and Wilkie, and also beyond this work, will be presented, 
in relation to several questions raised by Yomdin. The non-archimedean case is joint work with Forey and Loeser. The real case is joint work with Pila and Wilkie, continued by my PhD student S. Van Hille.  Some work with Binyamini and Novikov in the non-archimedean context will also be mentioned. The relations with questions by Yomdin is joint work with Friedland and Yomdin.