Hironaka's resolution of singularities is a celebrated discipline that has many applications in many areas of Mathematics. In this work we approach the desingularization of semialgebraic sets ${\mathcal S}\subset{\mathbb R}^n$. We prove the counterpart to `Hironaka's desingularization of real algebraic sets' in the semialgebraic setting. It involves Nash manifolds with corners instead of non-singular real algebraic sets. The obtained results are optimal when ${\mathcal S}$ is a closed semialgebraic set. The main tools involved are: Hironaka's desingularization of real algebraic sets, Hironaka's embedded desingularization of real algebraic subsets of non-singular real algebraic sets and drilling blow-ups of Nash manifolds with centers closed Nash submanifolds. We show how to `build' a Nash manifold with corners ${\mathcal Q}\subset{\mathbb R}^n$ from a suitable Nash manifold $M\subset{\mathbb R}^n$ (of its same dimension), which contains ${\mathcal Q}$ as a closed subset, by folding $M$ along the irreducible components of a normal-crossings divisor of $M$, which is the smallest Nash subset of $M$ that contains the boundary $\partial{\mathcal Q}$ of ${\mathcal Q}$. One can choose as the Nash manifold $M$ the Nash `double' $D({\mathcal Q})$ of ${\mathcal Q}$, which is analogous to the Nash double of a Nash manifold with boundary, but $D({\mathcal Q})$ takes into account the peculiarities of the boundary of a Nash manifold with corners. We include several applications: (1) Weak desingularization of closed semialgebraic sets using Nash manifolds with boundary, (2) Representation of compact semialgebraic sets connected by analytic paths as images under Nash maps of closed unit balls, (3) Construction of Nash models for compact orientable smooth surfaces of genus $g\geq0$, and (4) Nash approximation of continuous semialgebraic maps whose target spaces are Nash manifolds with corners.PDF: Desingularization of semialgebraic sets. Applications.pdf
