In this paper, a random graph process {$G$($t$)}_{$t$≥1}is studied and its degree sequence is analyzed. Let {$W$_{$t$}}_{$t$≥1}be an i.i.d. sequence. The graph process is defined so that, at each integer time$t$, a new vertex with$W$_{$t$}edges attached to it, is added to the graph. The new edges added at time$t$are then preferentially connected to older vertices, i.e., conditionally on$G$($t$-1), the probability that a given edge of vertex$t$is connected to vertex$i$is proportional to$d$_{$i$}($t$-1)+δ, where$d$_{$i$}($t$-1) is the degree of vertex$i$at time$t$-1, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent τ=min{τ_{W},τ_{P}}, where τ_{W}is the power-law exponent of the initial degrees {$W$_{$t$}}_{$t$≥1}and τ_{P}the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze.

We study the geometry of Hilbert schemes of points on abelian surfaces and Beauville’s generalized Kummer varieties in positive characteristics. The main result is that, in characteristic two, the addition map from the Hilbert scheme of two points to the abelian surface is a quasifibration such that all fibers are nonsmooth. In particular, the corresponding generalized Kummer surface is nonsmooth, and minimally elliptic singularities occur in the supersingular case. We unravel the structure of the singularities in dependence of$p$-rank and$a$-number of the abelian surface. To do so, we establish a McKay Correspondence for Artin’s wild involutions on surfaces. Along the line, we find examples of canonical singularities that are not rational singularities.

We prove that for the cases $X=\mathcal{C}(K)$ ($K$infinite) and$X$=$L$_{1}(μ) (μ σ-finite and atomless) it holds that every slice of the unit ball of the$N$-fold symmetric tensor product of$X$has diameter two. In fact, we prove more general results for weak neighborhoods relative to the unit ball. As a consequence, we deduce that the spaces of$N$-homogeneous polynomials on those classical Banach spaces have no points of Fréchet differentiability.

We prove the following extension of the Wiener–Wintner theorem and the Carleson theorem on pointwise convergence of Fourier series: For all measure-preserving flows ($X$,μ,$T$_{$t$}) and$f$∈$L$^{$p$}($X$,μ), there is a set$X$_{$f$}⊂$X$of probability one, so that for all$x$∈$X$_{$f$}, $$\lim_{s\downarrow0}\int_{s<|t|<1/s}e^{i\theta t} f(\textup{T}_tx)\,\frac{dt}t\quad\text{exists for all}\ \theta.$$ The proof is by way of establishing an appropriate oscillation inequality which is itself an extension of Carleson’s theorem.

We investigate whether a Stein manifold$M$which allows proper holomorphic embedding into ℂ^{$n$}can be embedded in such a way that the image contains a given discrete set of points and in addition follow arbitrarily close to prescribed tangent directions in a neighbourhood of the discrete set. The infinitesimal version was proven by Forstnerič to be always possible. We give a general positive answer if the dimension of$M$is smaller than$n$/2 and construct counterexamples for all other dimensional relations. The obstruction we use in these examples is a certain hyperbolicity condition.