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We exploit the Galois symmetries of the little disks operads to show that many differentials in the Goodwillie–Weiss spectral sequences approximating the homology and homotopy of knot spaces vanish at a prime 
$p$. Combined with recent results on the relationship between embedding calculus and finite-type theory, we deduce that the 
$(n+1)$th Goodwillie–Weiss approximation is a 
$p$-local universal Vassiliev invariant of degree 
$\leq n$ for every 
$n \leq p + 1$.
Let 
$G$ be a split semisimple algebraic group over a field and let 
$A^*$ be an oriented cohomology theory in the Levine–Morel sense. We provide a uniform approach to the 
$A^*$-motives of geometrically cellular smooth projective 
$G$-varieties based on the Hopf algebra structure of 
$A^*(G)$. Using this approach, we provide various applications to the structure of motives of twisted flag varieties.
We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over 
$\mathbb {Q}$ and 
$\mathbb {F}_q(t)$, and conclude with a pair of hyperkähler 4-folds over 
$\mathbb {Q}$. The latter is independently interesting as a new example of a transcendental Brauer–Manin obstruction to the Hasse principle. The source code for the various computations is supplied as supplementary material with the online version of this article.
$K(1)$-local 
$\mathrm {TR}$
We discuss some general properties of 
$\mathrm {TR}$ and its 
$K(1)$-localization. We prove that after 
$K(1)$-localization, 
$\mathrm {TR}$ of 
$H\mathbb {Z}$-algebras is a truncating invariant in the Land–Tamme sense, and deduce 
$h$-descent results. We show that for regular rings in mixed characteristic, 
$\mathrm {TR}$ is asymptotically 
$K(1)$-local, extending results of Hesselholt and Madsen. As an application of these methods and recent advances in the theory of cyclotomic spectra, we construct an analog of Thomason's spectral sequence relating 
$K(1)$-local 
$K$-theory and étale cohomology for 
$K(1)$-local 
$\mathrm {TR}$.