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We determine when the integral homology of the classifying space of a $PU(n)$
-gauge group over the sphere $S^2$
has torsion.
Bogomolov and Tschinkel [‘Algebraic varieties over small fields’, Diophantine Geometry, U. Zannier (ed.), CRM Series, 4 (Scuola Normale Superiore di Pisa, Pisa, 2007), 73–91] proved that, given two complex elliptic curves 
$E_1$ and 
$E_2$ along with even degree-
$2$ maps 
$\pi _j\colon E_j\to \mathbb {P}^1$ having different branch loci, the intersection of the image of the torsion points of 
$E_1$ and 
$E_2$ under their respective 
$\pi _j$ is finite. They conjectured (also in works with Fu) that the cardinality of this intersection is uniformly bounded independently of the elliptic curves. The recent proof of the uniform Manin–Mumford conjecture implies a full solution of the Bogomolov–Fu–Tschinkel conjecture. In this paper, we prove a generalisation of the Bogomolov–Fu–Tschinkel conjecture whereby, instead of even degree-
$2$ maps, one can use any rational functions of bounded degree on the elliptic curves as long as they have different branch loci. Our approach combines Nevanlinna theory with the uniform Manin–Mumford conjecture. With similar techniques, we also prove a result on lower bounds for ranks of elliptic curves over number fields.
We study curve-shortening flow for twisted curves in 
$\mathbb {R}^3$ (that is, curves with nowhere vanishing curvature 
$\kappa $ and torsion 
$\tau $) and define a notion of torsion-curvature entropy. Using this functional, we show that either the curve develops an inflection point or the eventual singularity is highly irregular (and likely impossible). In particular, it must be a Type-II singularity which admits sequences along which 
${\tau }/{\kappa ^2} \to \infty $. This contrasts strongly with Altschuler’s planarity theorem, which shows that 
${\tau }/{\kappa } \to 0$ along any essential blow-up sequence.
Our main purpose is to extend several results of interest that have been proved for modules over integral domains to modules over arbitrary commutative rings 
$R$ with identity. The classical ring of quotients 
$Q$ of 
$R$ will play the role of the field of quotients when zero-divisors are present. After discussing torsion-freeness and divisibility (Sections 2–3), we study Matlis-cotorsion modules and their roles in two category equivalences (Sections 4–5). These equivalences are established via the same functors as in the domain case, but instead of injective direct sums 
$\oplus Q$ one has to take the full subcategory of 
$Q$-modules into consideration. Finally, we prove results on Matlis rings, i.e. on rings for which 
$Q$ has projective dimension 
$1$ (Theorem 6.4).
