Thomason’s étale descent theorem for Bott periodic algebraic $K$–theory is generalized to any $MGL$ module over a regular Noetherian scheme of finite dimension. Over arbitrary Noetherian schemes of finite dimension, this generalizes the analogue of Thomason’s theorem for Weibel’s homotopy $K$–theory. This is achieved by amplifying the effects from the case of motivic cohomology, using the slice spectral sequence in the case of the universal example of algebraic cobordism. We also obtain integral versions of these statements: Bousfield localization at étale motivic cohomology is the universal way to impose étale descent for these theories. As applications, we describe the étale local objects in modules over these spectra and show that they satisfy the full six functor formalism, construct an étale descent spectral sequence converging to Bott-inverted motivic Landweber exact theories, and prove cellularity and effectivity of the étale versions of these motivic spectra.

We define a deformation of the triply graded Khovanov–Rozansky homology of a link $L$ depending on a choice of parameters $y_c$ for each component of $L$, which satisfies link-splitting properties similar to the Batson–Seed invariant. Keeping the $y_c$ as formal variables yields a link homology valued in triply graded modules over $\mathbb{Q}{[x_c,y_c]}_{c\in {\pi }_0(L)}$. We conjecture that this invariant restores the missing $Q\leftrightarrow T{Q}^{-1}$ symmetry of the triply graded Khovanov–Rozansky homology, and in addition satisfies a number of predictions coming from a conjectural connection with Hilbert schemes of points in the plane. We compute this invariant for all positive powers of the full twist and match it to the family of ideals appearing in Haiman’s description of the isospectral Hilbert scheme.

Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental group of the surface. For surfaces of genus at least two, such orbits correspond to homomorphisms with finite image. For genus one, they correspond to the finite or special dihedral representations. We also obtain an analogous result for bounded orbits in the moduli space.

Since its first use by Behrend, Bryan, and Szendrői in the computation of motivic Donaldson–Thomas (DT) invariants of ${\mathbb{𝔸}}_{ℂ}^3$, dimensional reduction has proved to be a crucial tool in motivic and cohomological DT theory. Inspired by a conjecture of Cazzaniga, Morrison, Pym and Szendrői on motivic DT invariants, work of Dobrovolska, Ginzburg and Travkin on exponential sums, and work of Orlov and Hirano on equivalences of categories of singularities, we generalize the dimensional reduction theorem in motivic and cohomological DT theory and use it to prove versions of the Cazzaniga–Morrison–Pym–Szendrői conjecture in these settings.

In this work we study the failure of the HKR theorem over rings of positive and mixed characteristic. For this we construct a filtered circle interpolating between the usual topological circle and a formal version of it. By mapping to schemes we produce this way an interpolation, realized in practice by the existence of a natural filtration, from Hochschild and (a filtered version of) cyclic homology to derived de Rham cohomology. In particular, we show that this recovers the filtration of Antieau (Ann. $K$–Theory 4 (2019) 505–519) and Bhatt, Morrow and Scholze (Publ. Math. Inst. Hautes Études Sci. 129 (2019) 199–310). The construction of our filtered circle is based on the theory of affine stacks and affinization introduced by the third author, together with some facts about schemes of Witt vectors.

We exhibit many examples of closed complex surfaces whose diffeomorphism groups are not simply connected and which contain loops that are not homotopic to loops of symplectomorphisms.

We show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, using the fact that they admit periodic, integrally minimal foliations by homogeneous hypersurfaces. For the geometric flow induced by the orbit–Einstein condition, we construct a Lyapunov function based on curvature estimates which come from real GIT.