We show that the additive higher Chow groups of regular schemes over a field induce a Zariski sheaf of pro-differential graded algebras, the Milnor range of which is isomorphic to the Zariski sheaf of big de Rham–Witt complexes. This provides an explicit cycle-theoretic description of the big de Rham–Witt sheaves. Several applications are derived.

For a Coxeter system and a representation $V$ of this Coxeter system, Soergel defined a category which is now called the category of Soergel bimodules and proved that this gives a categorification of the Hecke algebra when $V$ is reflection faithful. Elias and Williamson defined another category when $V$ is not reflection faithful and proved that this category is equivalent to the category of Soergel bimodules when $V$ is reflection faithful. Moreover, they proved the categorification theorem for their category with fewer assumptions on $V$. In this paper, we give a bimodule description of the Elias–Williamson category and re-prove the categorification theorem.

We give many examples of algebraic actions which are factors of Bernoulli shifts. These include certain harmonic models over left-orderable groups of large enough growth, as well as algebraic actions associated to certain lopsided elements in any left-orderable group. For many of our examples, the acting group is amenable so these actions are Bernoulli (and not just a factor of a Bernoulli), but there is no obvious Bernoulli partition.

We establish some cohomological bounds in $D$-module theory that are known in the holonomic case and folklore in general. The method rests on a generalization of the $b$-function lemma for non-holonomic $D$-modules.

Let $G$ be a split connected reductive group over a finite field of characteristic $p > 2$ such that $G_\text {der}$ is absolutely almost simple. We give a geometric construction of perverse $\mathbb {F}_p$-sheaves on the Iwahori affine flag variety of $G$ which are central with respect to the convolution product. We deduce an explicit formula for an isomorphism from the spherical mod $p$ Hecke algebra to the center of the Iwahori mod $p$ Hecke algebra. We also give a formula for the central integral Bernstein elements in the Iwahori mod $p$ Hecke algebra. To accomplish these goals we construct a nearby cycles functor for perverse $\mathbb {F}_p$-sheaves and we use Frobenius splitting techniques to prove some properties of this functor. We also prove that certain equal characteristic analogues of local models of Shimura varieties are strongly $F$-regular, and hence they are $F$-rational and have pseudo-rational singularities.

The trace of the $n$-framed surgery on a knot in $S^{3}$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded $2$-sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable $3$-dimensional knot invariants. For each $n$, this provides conditions that imply a knot is topologically $n$-shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.

We prove that a formula for the ‘pluricanonical’ double ramification cycle proposed by Janda, Pandharipande, Pixton, Zvonkine, and the second-named author is in fact the class of a cycle constructed geometrically by the first-named author. Our proof proceeds by a detailed explicit analysis of the deformation theory of the double ramification cycle, both to first and to higher order.

We give a corrected version of Theorem 3, Lemma 4, and Proposition 9 in the above-mentioned paper, which are incorrect as stated (as was pointed out by O. Gabber).