• Baire category
• Convex integration

In this thesis we present two techniques that are useful in the construction of bizarre examples and counterexamples in the fields of analysis and geometry.

In the first chapter, we provide some classical examples that exploit the Baire category theorem. In particular, we prove the existence of a bounded 1/2-Holder function such that it is not Lipschitz continuous in any sub-interval, or the residuality of Lipschitz functions with constant = 1 in any sub-interval in the space of Lipschitz functions with constant <= 1. Moreover, in some case, we explain how the same ingredients of a "Baire proof" can be used in order to construct an explicit solution. Then, we prove that in the space of functions f in L^2(a,b) with |f(x)| <= 1 a.e., the set of functions such that f(x) in {-1,1} a.e. is residual with respect to the metric that induces the weak convergence in L^2(a,b).

In the next three chapters, we discuss three applications of the method which is now known as Convex Integration.

In Chapter 2, we show the density of curves with unit norm derivative into the set of curves with derivative of norm less than 1. This set is also residual with respect to a suitable metric.

In Chapter 3, we show that there exist infinitely many bounded u: R^3 -> R^3 such that div u=0 in the sense of distributions and |u(x)|=1 on an open and bounded set and 0 otherwise.

In Chapter 4, we consider a special case of the seminal Nash-Kuiper Theorem. In particular, if we let D^2 be the 2-dimensional disk and N >= 3, we apply the convex integration procedure in order to prove that the set of C^1 isometric immersions from D^2 to R^N is dense (with respect to the C^0 norm) in the set of smooth short immersions from D^2 to R^N.