We consider two categorifications of the cohomology of a topological space X by taking coefficients in the category of differential graded categories. We consider both derived global sections of a constant presheaf and singular cohomology and find the resulting dg-categories are quasi-equivalent and moreover quasi-equivalent to representations in perfect complexes of chains on the loop space of X.

We identify Morita cohomology, which is a categorification of the cohmology of a topological space X, with the category of homotopy locally constant sheaves of perfect complexes on X.

We introduce bow varieties and construct some ALF spaces as bow varieties.

In this paper we study the images of the Johnson homomorphisms of the basis-conjugating automorphism groups of free groups and free metabelian groups. In particular, we show that the Johnson image is contained in a certain proper Lie subalgebra $\mathfrak{p}$Mn of the derivation algebra of the Chen Lie algebra. Furthermore, we completely determine the Johnson images, and give the abelianisation of $\mathfrak{p}$Mn as a Lie algebra by using Morita's trace maps.

We prove that the coefficients of the mock theta functions

\begin{eqnarray*} f(q) = \sum_{n=1}^{\infty} \frac{ q^{n^2}}{(1+q)^2 (1+q^2)^2 \cdots (1+q^n)^2 } \end{eqnarray*}

\begin{eqnarray*} \omega(q)=1+\sum_{n=1}^\infty \frac{q^{2n^2+2n}}{(1+q)^2(1+q^3)^2\cdots (1+q^{2n+1})^2} \end{eqnarray*}

Let k be a number field and T a k-torus. Consider a family of torsors under T, i.e. a morphism f : X → ℙ1k from a projective, smooth k-variety X to ℙ1k, the generic fibre Xη → η of which is a smooth compactification of a principal homogeneous space under T ⊗k η. We study the Brauer–Manin obstruction to the Hasse principle and to weak approximation for X, assuming Schinzel's hypothesis. We generalise Wei's recent results [21]. Our results are unconditional if k = Q and all non-split fibres of f are defined over Q. We also establish an unconditional analogue of our main result for zero-cycles of degree 1.

We give a short proof of a slightly weaker version of the multilinear Kakeya inequality proven by Bennett, Carbery and Tao.

I propose a definition of left/right connection along a strong homotopy Lie–Rinehart algebra. This allows me to generalise simultaneously representations up to homotopy of Lie algebroids and actions of L∞ algebras on graded manifolds. I also discuss the Schouten-Nijenhuis calculus associated to strong homotopy Lie–Rinehart connections.