The trace (or zeroth Hochschild homology) of Khovanov’s Heisenberg category is identified with a quotient of the algebra $W_{1+\infty }$. This induces an action of $W_{1+\infty }$ on the center of the categorified Fock space representation, which can be identified with the action of $W_{1+\infty }$ on symmetric functions.

Let $F$ be a unramified finite extension of $\mathbb{Q}_{p}$ and $\overline{\unicode[STIX]{x1D70C}}$ be an irreducible mod $p$ two-dimensional representation of the absolute Galois group of $F$. The aim of this article is the explicit computation of the Kisin variety parameterizing the Breuil–Kisin modules associated to certain families of potentially Barsotti–Tate deformations of $\overline{\unicode[STIX]{x1D70C}}$. We prove that this variety is a finite union of products of $\mathbb{P}^{1}$. Moreover, it appears as an explicit closed connected subvariety of $(\mathbb{P}^{1})^{[F:\mathbb{Q}_{p}]}$. We define a stratification of the Kisin variety by locally closed subschemes and explain how the Kisin variety equipped with its stratification may help in determining the ring of Barsotti–Tate deformations of $\overline{\unicode[STIX]{x1D70C}}$.

We study actions of Lie supergroups, in particular, the hitherto elusive notion of orbits through odd (or more general) points. Following categorical principles, we derive a conceptual framework for their treatment and therein prove general existence theorems for the isotropy (or stabiliser) supergroups and orbits through general points. In this setting, we show that the coadjoint orbits always admit a (relative) supersymplectic structure of Kirillov–Kostant–Souriau type. Applying a family version of Kirillov’s orbit method, we decompose the regular representation of an odd Abelian supergroup into an odd direct integral of characters and construct universal families of representations, parametrised by a supermanifold, for two different super variants of the Heisenberg group.

The purpose of this paper is to provide a large class of initial data which generates global smooth solution of the 3D inhomogeneous incompressible Navier–Stokes system in the whole space $\mathbb{R}^{3}$. This class of data is based on functions which vary slowly in one direction. The idea is that 2D inhomogeneous Navier–Stokes system with large data is globally well-posed and we construct the 3D approximate solutions by the 2D solutions with a parameter. One of the key point of this study is the investigation of the time decay properties of the solutions to the 2D inhomogeneous Navier–Stokes system. We obtained the same optimal decay estimates as the solutions of 2D homogeneous Navier–Stokes system.

We introduce the open degree of a compact space, and we show that for every natural number $n$, the separable Rosenthal compact spaces of degree $n$ have a finite basis.

We determine the Galois representations inside the $\ell$-adic cohomology of some unitary Shimura varieties at split places where they admit uniformization by finite products of Drinfeld upper half spaces. Our main results confirm Langlands–Kottwitz’s description of the cohomology of Shimura varieties in new cases.