We show that the category of ind-coherent sheaves on a quasi-smooth scheme is naturally tensored over the category of sheared D-modules on its shifted cotangent bundle, commuting with its natural action of categorified Hoschschild cochains. We prove that it defines a Morita equivalence as such. We then extend these results to quasi-smooth Artin stacks. As a consequence of our formalism, we are able to articulate a precise sense in which the space of unramified automorphic functions over a function field localizes over the stack of arithmetic Arthur parameters.

We determine the A(1)-homotopy of the topological cyclic homology of the connective real K-theory spectrum ko. The answer has an associated graded that is a free $\mathbb{F}_2[v_2^4]$-module of rank 52, on explicit generators in stems $-1 \leqslant \ast \leqslant 30$. The calculation is achieved by using prismatic and syntomic cohomology of ko as introduced by Hahn, Raksit and Wilson, extending work of Bhatt, Morrow and Scholze from the case of classical commutative rings to $\mathbb{E}_\infty$-rings. A new feature in our case is that there are nonzero differentials in the motivic spectral sequence from syntomic cohomology to topological cyclic homology.

We study the relationship between higher Du Bois singularities and K-regularity, a notion that measures the $\mathbb A^1$-invariance of the algebraic K-groups. Building on this relationship, we establish a strengthened form of Vorst’s conjecture for local complete intersections in characteristic zero. Our work also provides tools to construct new examples that illustrate various phenomena in the study of K-regularity. The main inputs for our results are vanishing theorems for the Du Bois complexes.

The landmark quantum complexity result MIP$^*$ $=$ RE has been used to prove the existence of a non-Connes-embeddable tracial von Neumann algebra. Recently, similar ideas have been used to give a negative solution to the Aldous–Lyons conjecture: there is a non-co-sofic invariant random subgroup (IRS) on any non-abelian free group. We define a notion of hyperlinearity for an IRS and show that there is a non-co-hyperlinear IRS on any non-abelian free group. As a corollary, we prove that there is a relation whose von Neumann algebra is not Connes embeddable. We do this by significantly simplifying the reduction of Aldous–Lyons to non-local games, removing the need for subgroup tests entirely.

We prove that all p-adic period domains (and their non-minuscule analogues) are geometrically connected. This answers a question of Hartl and has consequences for the geometry of Shimura and local Shimura varieties.

We correct the proof of Corollary 6.21 in the paper The p-cyclic McKay correspondence via motivic integration [Compositio Math. 150 (2014), 1125–1168].