Let ${\mathcal A}$ be a line arrangement in the complex projective plane $\mathds{C}\mathds{P}^2$. We define and describe the inclusion map of the boundary manifold, the boundary of a closed regular neighbourhood of ${\mathcal A}$, in the exterior of the arrangement. We obtain two explicit descriptions of the map induced on the fundamental groups. These computations provide a new minimal presentation of the fundamental group of the complement.

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. Putman and the authors proved that this group is generated by Dehn twists about separating curves fixed by the hyperelliptic involution. In this paper, we introduce an algorithmic approach to factoring a wide class of elements of the hyperelliptic Torelli group into such Dehn twists, and apply our methods to several basic types of elements. As one consequence, we answer an old question of Dennis Johnson.

On a (pseudo-)Riemannian manifold (${\mathcal M}$, g), some fields of endomorphisms i.e. sections of End(T${\mathcal M}$) may be parallel for g. They form an associative algebra $\mathfrak e$, which is also the commutant of the holonomy group of g. As any associative algebra, $\mathfrak e$ is the sum of its radical and of a semi-simple algebra $\mathfrak s$. Here we study $\mathfrak s$: it may be of eight different types, including the generic type $\mathfrak s$ = ${\mathbb R}$ Id, and the Kähler and hyperkähler types $\mathfrak s$ ≃ ${\mathbb C}$ and $\mathfrak s$ ≃ ${\mathbb H}$. This is a result on real, semi-simple algebras with involution. For each type, the corresponding set of germs of metrics is non-empty; we parametrize it. We give the constraints imposed to the Ricci curvature by parallel endomorphism fields.

Given any positive integer M, we show that there are infinitely many real quadratic fields that do not admit universal quadratic forms with even cross coefficients in M variables.

For every complex polynomial p(z), closed point sets are defined, called loci of p(z). A closed set Ω ⊆ ${\mathbb C}$* is a locus of p(z) if it contains a zero of any of its apolar polynomials and is the smallest such set with respect to inclusion. Using the notion locus, some classical theorems in the geometry of polynomials can be refined. We show that each locus is a Lebesgue measurable set and investigate its intriguing connections with the higher-order polar derivatives of p.

The ring A = $\mathbb{F}$r[T] and its fraction field k, where r is a power of a prime p, are considered as analogues of the integers and rational numbers respectively. Let K/k be a finite extension and let φ be a Drinfeld A-module over K of rank d and Γ ⊂ K be a finitely generated free A-submodule of K, the A-module structure coming from the action of φ. We consider the problem of determining the number of primes ℘ of K for which the reduction of Γ modulo ℘ is equal to $\mathbb{F}$℘ (the residue field of the prime ℘). We can show that there is a natural density of primes ℘ for which Γ mod ℘ is equal to $\mathbb{F}$℘. In certain cases, this density can be seen to be positive.

Let Ω0 be a polygon in $\mathbb{R}$2, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that Ωε is a family of surfaces with ${\mathcal C}$∞ boundary which converges to Ω0 smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov [6], Kac [8] and McKean–Singer [13] recognised that certain heat trace coefficients, in particular the coefficient of t0, are not continuous as ε ↘ 0. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domain Z which models the corner formation. The result applies to both Dirichlet and Neumann boundary conditions. We also include a discussion of what one might expect in higher dimensions.

Tian initiated the study of incomplete Kähler–Einstein metrics on quasi–projective varieties with cone-edge type singularities along a divisor, described by the cone-angle 2π(1-α) for α∈ (0, 1). In this paper we study how the existence of such Kähler–Einstein metrics depends on α. We show that in the negative scalar curvature case, if such Kähler–Einstein metrics exist for all small cone-angles then they exist for every α∈((n+1)/(n+2), 1), where n is the dimension. We also give a characterisation of the pairs that admit negatively curved cone-edge Kähler–Einstein metrics with cone angle close to 2π. Again if these metrics exist for all cone-angles close to 2π, then they exist in a uniform interval of angles depending on the dimension only. Finally, we show how in the positive scalar curvature case the existence of such uniform bounds is obstructed.

Let $\mathcal{P}$ be a partition of a finite set X. We say that a transformation f : X → X preserves (or stabilises) the partition $\mathcal{P}$ if for all P ∈ $\mathcal{P}$ there exists Q ∈ $\mathcal{P}$ such that Pf ⊆ Q. Let T(X, $\mathcal{P}$) denote the semigroup of all full transformations of X that preserve the partition $\mathcal{P}$.

In 2005 Pei Huisheng found an upper bound for the minimum size of the generating sets of T(X, $\mathcal{P}$), when $\mathcal{P}$ is a partition in which all of its parts have the same size. In addition, Pei Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to solve Pei Huisheng's conjecture.

The aim of this paper is to solve the more complex problem of finding the minimum size of the generating sets of T(X, $\mathcal{P}$), when $\mathcal{P}$ is an arbitrary partition. Again we use representation theory to find the minimum number of elements needed to generate the wreath product of finitely many symmetric groups, and then use this result to solve the problem.

The paper ends with a number of problems for experts in group and semigroup theories.

Let R be a standard graded algebra over a field k. We prove an Auslander–Buchsbaum formula for the absolute Castelnuovo–Mumford regularity, extending important cases of previous works of Chardin and Römer. For a bounded complex of finitely generated graded R-modules L, we prove the equality reg L = maxi ∈$_{\mathbb Z}$ {reg Hi(L) − i} given the condition depth Hi(L) ⩾ dim Hi+1(L) - 1 for all i < sup L. As applications, we recover previous bounds on regularity of Tor due to Caviglia, Eisenbud–Huneke–Ulrich, among others. We also obtain strengthened results on regularity bounds for Ext and for the quotient by a linear form of a module.