The paper considers three questions about independent random points uniformly distributed in isotropic symmetric convex bodies $K, T_1,\ldots, T_s$. (a) Let $\varepsilon\,{\in}\, (0,1)$ and let $x_1,\ldots, x_N$ be chosen from K. Is it true that if $N\,{\geq}\, C(\varepsilon )n\log n$, then \[\left\| I-\frac{1}{NL_K^2}\sum_{i=1}^Nx_i\otimes x_i\right\|<\varepsilon\] with probability greater than $1\,{-}\,\varepsilon $? (b) Let $x_i$ be chosen from $T_i$. Is it true that the unconditional norm \[\|{\bf t}\|=\int_{T_1}\!{\ldots}\int_{T_s}\left\|\sum_{i=1}^st_ix_i\right\|_K\,dx_s\ldots dx_1\] is well comparable to the Euclidean norm in ${\mathbb R}^s$? (c) Let $x_1,\ldots, x_N$ be chosen from K. Let ${\mathbb E}\,(K,N):={\mathbb E}\,|{\rm conv}\{ x_1,\ldots, x_N\}|^{1/n}$ be the expected volume radius of their convex hull. Is it true that ${\mathbb E}\,(K,N)\,{\simeq}\, {\mathbb E}\,(B(n),N)$ for all N, where $B(n)$ is the Euclidean ball of volume 1?

It is proved that the answers to these questions are affirmative if there is a restriction to the class of unconditional convex bodies. The main tools come from recent work of Bobkov and Nazarov. Some observations about the general case are also included.

it is shown that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups induces an infinite loop map on the classifying spaces of the stable groups after plus construction. the proof uses automorphisms of free groups with boundaries which play the role of mapping class groups of surfaces with several boundary components.

Necessary and sufficient conditions are given for linearly recurrent Cantor dynamical systems to have measurable and continuous eigenfunctions. Also an example of a linearly recurrent system with a nontrivial Kronecker factor and a trivial maximal equicontinuous factor is constructed explicitly.

Let $\{ \vf_k(x),\,k=1,2,\ldots\}$ be an arbitrary orthonormal system on [0,1] that is uniformly bounded by a constant $M$. Let $T$ be a subset of [0,1]$^2$ such that the Fourier series of all Lebesgue integrable functions on [0,1]$^2$ with respect to the product system $\{ \vf_k(x) \vf_l(y),\,k, l=1,2,\ldots\}$ converge in measure by squares on $T$. The following problem is studied. How large may the measure of $T$ be?

A theorem is proved that implies that for each such system, there is $\mu_2T \leq 1-M^{-4} $ (for the $d$-fold product systems, $\mu_dT \,{\leq}\, 1-M^{-2d}$, d\,{\geq}\, 2$). This estimate is sharp in the class of all such product systems.

given an $n$-dimensional compact closed oriented manifold $m$ and a field $\mathbb k$, f. cohen and l. taylor have constructed a spectral sequence, ${\mathcal e}(m,n,\mathbb k)$, converging to the cohomology of the space of ordered configurations of $n$ points in $m$. the symmetric group $\sigma_n$ acts on this spectral sequence giving a spectral sequence of $\sigma_n$-differential graded commutative algebras. here, an explicit description is provided of the invariants algebra $(e_1,d_1)^{\sigma_n}$ of the first term of $\mathcal e(m,n,\mathbb q)$. this determination is applied in two directions.

(a) in the case of a complex projective manifold or of an odd-dimensional manifold $m$, the cohomology algebra $h^*(c_n(m);\mathbb q)$ of the space of unordered configurations of $n$ points in $m$ is obtained (the concrete example of $p^2(\mathbb c)$ is detailed).

(b) the degeneration of the spectral sequence formed of the $\sigma_n$-invariants $\mathcal e(m,n,\mathbb q)^{\sigma_n}$ at level 2 is proved for any manifold $m$.

these results use a transfer map and are also true with coefficients in a finite field $\mathbb f_p$ with $p>n$.

The paper shows the existence of several new families of noncompact constant mean curvature surfaces: (i) singly punctured surfaces of arbitrary genus $g\,{\geq}\,1$, (ii) doubly punctured tori, and (iii) doubly periodic surfaces with Delaunay ends.