For each of the groups G = O(2), SU(2), U(2), we compute the integral and $\mathbb{F}_2$-cohomology rings of BcomG (the classifying space for commutativity of G), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy type of EcomG (the homotopy fiber of the inclusion BcomG → BG), and some low-dimensional homotopy groups of BcomG.

We study generating functions of ordinary and plane partitions coloured by the action of a finite subgroup of the corresponding special linear group. After reviewing known results for the case of ordinary partitions, we formulate a conjecture concerning a basic factorisation property of the generating function of coloured plane partitions that can be thought of as an orbifold analogue of a conjecture of Maulik et al., now a theorem, in three-dimensional Donaldson–Thomas theory. We study natural quantisations of the generating functions arising from geometry, discuss a quantised version of our conjecture, and prove a positivity result for the quantised coloured plane partition function under a geometric assumption.

We study C1-robustly transitive and nonhyperbolic diffeomorphisms having a partially hyperbolic splitting with one-dimensional central bundle whose strong un-/stable foliations are both minimal. In dimension 3, an important class of examples of such systems is given by those with a simple closed periodic curve tangent to the central bundle. We prove that there is a C1-open and dense subset of such diffeomorphisms such that every nonhyperbolic ergodic measure (i.e. with zero central exponent) can be approximated in the weak* topology and in entropy by measures supported in basic sets with positive (negative) central Lyapunov exponent. Our method also allows to show how entropy changes across measures with central Lyapunov exponent close to zero. We also prove that any nonhyperbolic ergodic measure is in the intersection of the convex hulls of the measures with positive central exponent and with negative central exponent.

Given a positive integer n let ω (n) denote the number of distinct prime factors of n, and let a be fixed positive integer. Extending work of Kubilius, we develop a bivariate probabilistic model to study the joint distribution of the deterministic vectors (ω(n), ω(n + a)), with n ≤ x as x → ∞, where n and n + a belong to a subset of ℕ with suitable properties. We thus establish a quantitative version of a bivariate analogue of the Erdős–Kac theorem on proper subsets of ℕ.

We give three applications of this result. First, if y = x0(1) is not too small then we prove (in a quantitative way) that the y-truncated Möbius function μy has small binary autocorrelations. This gives a new proof of a result due to Daboussi and Sarkőzy. Second, if μ(n; u) :=e(uω(n)), where u ∈ ℝ then we show that μ(.; u) also has small binary autocorrelations whenever u = o(1) and $u\sqrt {\mathop {\log }\nolimits_2 x} \to \infty$, as x → ∞. These can be viewed as partial results in the direction of a conjecture of Chowla on binary correlations of the Möbius function.

Our final application is related to a problem of Erdős and Mirsky on the number of consecutive integers less than x with the same number of divisors. If $y = x^{{1 \over \beta }}$, where β = β(x) satisfies certain mild growth conditions, we prove a lower bound for the number of consecutive integers n ≤ x that have the same number of y-smooth divisors. Our bound matches the order of magnitude of the one conjectured for the original Erdős-Mirsky problem.

Let ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 < p < ∞, has a positive answer.

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer k, we are interested in the set of all closed geodesics with at least k (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in k (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like k for growing k.

We show that provided that n is a power of two, any nD measures in ℝn can be bisected by an arrangement of D hyperplanes.