Cordian Riener Aalto University Aalto Science Institute Let R be a real closed field. We prove upper bounds on the equivariant Betti numbers of symmetric algebraic and semi-algebraic subsets of Rk. More precisely, we prove that if S⊂Rk is a semi-algebraic subset defined by a finite set of s symmetric polynomials of degree at most d, then the sum of the Sk equivariant Betti numbers of S with coefficients in Q is bounded by s5d(kd)O(d) . Unlike the well known classical bounds due to Oleinik and Petrovskii, Thom and Milnor on the Betti numbers of (possibly non-symmetric) real algebraic varieties and semi-algebraic sets, the above bound is polynomial in k when the degrees of the defining polynomials are bounded by a constant. Moreover, our bounds are asymptotically tight. As an application we improve the best known bound on the Betti numbers of the projection of a compact semi-algebraic set improving for any fixed degree the best previously known bound for this problem due to Gabrielov, Vorobjov and Zell. (Joint work with Saugata Basu)