**Seminario de Álgebra y Combinatoria**

**Viernes 1 de abril a las 12:00 en el aula 420 (módulo 17) **

**Conferenciante: José F. Fernando (Universidad Complutense de Madrid) **

**Título: On a solution to Shiota's Conjecture to characterize **

**Nash images of Euclidean spaces **

**Resumen: In this work we characterize the subsets of $R^n$ that are images of Nash maps $f:R^mtoR^n$. We prove Shiota's conjecture and show that em a subset $SssubsetR^n$ is the image of a Nash map $f:R^mtoR^n$ if and only if $Ss$ is semialgebraic, pure dimensional of dimension $dleq m$ and there exists an analytic path $alpha:[0,1]toSs$ whose image meets all the connected components of the set of regular points of $Ss$em. Given a semialgebraic set $SssubsetR^n$ satisfying the previous properties, we provide a theoretical strategy to construct (after Nash approximation) a Nash map whose image is the semialgebraic set $Ss$. This strategy includes resolution of singularities, relative Nash approximation on Nash manifolds with boundary and other tools (such as the drilling blow-up) constructed ad hoc for Nash manifolds and Nash subsets that may have further applications to approach new problems. **

**Some remarkable consequences are the following: (1) pure dimensional irreducible semialgebraic sets of dimension $d$ with arc-symmetric closure are Nash images of $R^d$; (2) semialgebraic sets are projections of irreducible algebraic sets whose connected components are Nash diffeomorphic to Euclidean spaces; and (3) compact $d$-dimensional smooth manifolds with boundary are smooth images of $R^d$. **