An asymptotic formula is established for the number of $\mathbb{Q}$-rational points of bounded height on a nonsingular quartic Del Pezzo surface with a conic bundle structure.

For any $1\le p\le \infty$ different from $2$, we give examples of noncommutative $L^p$-spaces without the completely bounded approximation property. Let $F$ be a nonarchimedian local field. If $p>4$ or $p<4/3$ and $r\ge 3$ these examples are the noncommutative $L^p$-spaces of the von Neumann algebra of lattices in $\mathrm{S}{\mathrm{L}}_r\left(F\right)$ or in $\mathrm{S}{\mathrm{L}}_r\left(\mathbb{R}\right)$. For other values of $p$ the examples are the noncommutative $L^p$-spaces of the von Neumann algebra of lattices in $\mathrm{S}{\mathrm{L}}_r\left(F\right)$ for $r$ large enough depending on $p$.

We also prove that if $r\ge 3$ lattices in $\mathrm{S}{\mathrm{L}}_r\left(F\right)$ or $\mathrm{S}{\mathrm{L}}_r\left(\mathbb{R}\right)$ do not have the approximation property of Haagerup and Kraus. This provides examples of exact $C^{\ast }$-algebras without the operator space approximation property.

We show that on an arbitrary, finitely generated, non-virtually-solvable linear group, any two independent random walks will eventually generate a free subgroup. In fact, this will hold for an exponential number of independent random walks.

There is a remarkable Drinfeld associator given by Kontsevich’s integrals over configuration spaces of points in the plane. Using this Drinfeld associator we show that Kontsevich’s and Tamarkin’s formalities of the little discs operad are homotopic. The basic technical tool for this result is an $L_{\infty }$-algebra of graphs whose cohomology is the Drinfeld-Kohno Lie algebra of infinitesimal pure braids.