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Grothendieck and Harder proved that every principal bundle over the projective line with split reductive structure group (and trivial over the generic point) can be reduced to a maximal torus. Furthermore, this reduction is unique modulo automorphisms and the Weyl group. In a series of six variations on this theme, we prove corresponding results for principal bundles over the following schemes and stacks: (1) a line modulo the group of nth roots of unity; (2) a football, that is, an orbifold of genus zero with two marked points; (3) a gerbe over a football whose structure group is the nth roots of unity; (4) a chain of lines meeting in nodes; (5) a line modulo an action of a split torus; and (6) a chain modulo an action of a split torus. We also prove that the automorphism groups of such bundles are smooth, affine, and connected.
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For every prime $p$, Mohan Kumar constructed examples of stably free modules of rank $p$ on suitable $(p+1)$-dimensional smooth affine varieties. This note discusses how to detect the corresponding unimodular rows in motivic cohomology. Using the recent developments in the $mathbb{A}^1$-obstruction classification of vector bundles, this provides an alternative proof of non-triviality of Mohan Kumars stably free modules. The reinterpretation of Mohan Kumars examples also allows to produce interesting examples of stably trivial torsors for other algebraic groups.
Several variants of the classic Fibonacci inflation tiling are considered in an illustrative fashion, in one and in two dimensions, with an eye on changes or robustness of diffraction and dynamical spectra. In one dimension, we consider extension mechanisms of deterministic and of stochastic nature, while we look at direct product variations in a planar extension. For the pure point part, we systematically employ a cocycle approach that is based on the underlying renormalisation structure. It allows explicit calculations, particularly in cases where one meets regular model sets with Rauzy fractals as windows.
In this note, we unify and extend various concepts in the area of $G$-complete reducibility, where $G$ is a reductive algebraic group. By results of Serre and Bate--Martin--R{o}hrle, the usual notion of $G$-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of $G$. We show that other variations of this notion, such as relative complete reducibility and $sigma$-complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.
Schinzel and Wojcik have shown that if $alpha, beta$ are rational numbers not $0$ or $pm 1$, then $mathrm{ord}_p(alpha)=mathrm{ord}_p(beta)$ for infinitely many primes $p$, where $mathrm{ord}_p(cdot)$ denotes the order in $mathbb{F}_p^{times}$. We begin by asking: When are there infinitely many primes $p$ with $mathrm{ord}_p(alpha) > mathrm{ord}_p(beta)$? We write down several families of pairs $alpha,beta$ for which we can prove this to be the case. In particular, we show this happens for 100% of pairs $A,2$, as $A$ runs through the positive integers. We end on a different note, proving a version of Schinzel and W{o}jciks theorem for the integers of an imaginary quadratic field $K$: If $alpha, beta in mathcal{O}_K$ are nonzero and neither is a root of unity, then there are infinitely many maximal ideals $P$ of $mathcal{O}_K$ for which $mathrm{ord}_P(alpha) = mathrm{ord}_P(beta)$.
We present a new approach to the Marcinkiewicz interpolation inequality for the distribution function of the Hilbert transform, and prove an abstract version of this inequality. The approach uses logarithmic determinants and new estimates of canonical products of genus one.
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