in this talk, we consider riemannian manifolds with almost non-negative scalar curvature and perelman entropy. we establish an $\epsilon$-regularity theorem showing that such a space must be close to euclidean space in a suitable sense. interestingly, such a result is false with respect to the gromov-hausdorff and intrinsic flat distances, and more generally the metric space structure is not controlled under entropy and scalar lower bounds. instead, we introduce the notion of the $d_p$ distance between (in particular) riemannian manifolds, which measures the distance between $w^{1,p}$ sobolev spaces, and it is with respect to this distance that the $\epsilon$ regularity theorem holds. we will discuss various applications to manifolds with scalar curvature and entropy lower bounds, including a compactness and limit structure theorem for sequences, a uniform $l^\infty$ sobolev embedding, and a priori $l^p$ scalar curvature bounds for $p<1$. \\ \\ this is joint work with man-chun lee and aaron naber.