Given a totally nonholonomic distribution of rank $2$ on a $3$-dimensional manifold, we investigate the size of the set of points that can be reached by singular horizontal paths starting from the same point. In this setting, by the Sard conjecture, that set should be a subset of the so-called Martinet surface of $2$-dimensional Hausdorff measure zero. We prove that the conjecture holds in the case where the Martinet surface is smooth. Moreover, we address the case of singular real-analytic Martinet surfaces, and we show that the result holds true under an assumption of nontransversality of the distribution on the singular set of the Martinet surface. Our methods rely on the control of the divergence of vector fields generating the trace of the distribution on the Martinet surface and some techniques of resolution of singularities.

Let $\Omega \subset {\mathbb{R}}^{n+1}$, $n\ge 1$, be a corkscrew domain with Ahlfors–David regular boundary. In this article we prove that $\partial \Omega$ is uniformly $n$-rectifiable if every bounded harmonic function on $\Omega$ is $\epsilon$-approximable or if every bounded harmonic function on $\Omega$ satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when $\Omega ={\mathbb{R}}^{n+1}\setminus E$ and $E$ is Ahlfors–David regular. Our results establish a conjecture posed by Hofmann, Martell, and Mayboroda, in which they proved the converse statements. Here we also obtain two additional criteria for uniform rectifiability, one in terms of the so-called $S<N$ estimates and another in terms of a suitable corona decomposition involving harmonic measure.

Given a $0$-connective motivic spectrum $E\in \mathbf{SH}\left(k\right)$ over a perfect field $k$, we determine ${\underline{h}}_0$ of the associated motive $ME\in \mathbf{DM}\left(k\right)$ in terms of ${\underline{\pi }}_0\left(E\right)$. Using this, we show that if $k$ has finite $2$-étale cohomological dimension, then the functor $M:\mathbf{SH}\left(k\right)\to \mathbf{DM}\left(k\right)$ is conservative when restricted to the subcategory of compact spectra and induces an injection on Picard groups. We extend the conservativity result to fields of finite virtual $2$-étale cohomological dimension by considering what we call real motives.

We compute the Hodge and the de Rham cohomology of the classifying space $\mathit{BG}$ (defined as étale cohomology on the algebraic stack $\mathit{BG}$) for reductive groups $G$ over many fields, including fields of small characteristic. These calculations have a direct relation with representation theory, yielding new results there. The calculations are closely analogous to, but not always the same as, the cohomology of classifying spaces in topology.