For a finite extension F of ${\mathbb Q}_p$ and $n \geq 1$, we show that the category of Lubin-Tate bundles on the $(n-1)$-dimensional Drinfeld symmetric space is equivalent to the category of finite-dimensional smooth representations of the group of units of the division algebra of invariant $1/n$ over F.

For a uniformly locally finite metric space $(X, d)$, we investigate coarse flows on its uniform Roe algebra $\mathrm {C}^*_u(X)$, defined as one-parameter groups of automorphisms whose differentiable elements include all partial isometries arising from partial translations on X. We first show that any flow $\sigma $ on $\mathrm {C}^*_u(X)$ corresponds to a (possibly unbounded) self-adjoint operator h on $\ell _2(X)$ such that $\sigma _t(a) = e^{ith} a e^{-ith}$ for all $t \in \mathbb {R}$, allowing us to focus on operators h that generate flows on $\mathrm {C}^*_u(X)$.

Assuming Yu’s property A, we prove that a self-adjoint operator h on $\ell _2(X)$ induces a coarse flow on $\mathrm {C}^*_u(X)$ if and only if h can be expressed as $h = a + d$, where $a \in \mathrm {C}^*_u(X)$ and d is a diagonal operator with entries forming a coarse function on X. We further study cocycle equivalence and cocycle perturbations of coarse flows, showing that, under property A, any coarse flow is a cocycle perturbation of a diagonal flow. Finally, for self-adjoint operators h and k that induce coarse flows on $\mathrm {C}^*_u(X)$, we characterize conditions under which the associated flows are either cocycle perturbations of each other or cocycle conjugate to each other. In particular, if $h - k$ is bounded, then the flow induced by h is a cocycle perturbation of the flow induced by k.

In recent years, b-symplectic manifolds have emerged as important objects in symplectic geometry. These manifolds are Poisson manifolds that exhibit symplectic behaviour away from a distinguished hypersurface, where the symplectic form degenerates in a controlled manner. Inspired by this rich landscape, E-structures were introduced by Nest and Tsygan in [NT01] as a comprehensive framework for exploring generalizations of b-structures. This paper initiates a deeper investigation into their Poisson facets, building on foundational work by [MS21]. We also examine the closely related concept of almost regular Poisson manifolds, as studied in [AZ17], which reveals a natural Poisson groupoid associated with these structures.

In this article, we investigate the intricate relationship between E-structures and almost regular Poisson structures. Our comparative analysis not only scrutinizes their Poisson properties but also offers explicit formulae for the Poisson structure on the Poisson groupoid associated to the E-structures as both Poisson manifolds and singular foliations. In doing so, we reveal an interesting link between the existence of commutative frames and Darboux-Carathéodory-type expressions for the relevant structures.

We establish several new properties of the p-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In particular, we reprove Scholze’s basic finiteness theorems, prove a duality theorem, and show a kind of partial Künneth formula. Using these results, we deduce bounds on Gelfand-Kirillov dimension, together with some new vanishing and nonvanishing results.

Our key new tool is the six functor formalism with solid almost $\mathcal {O}^+/p$-coefficients developed recently by the second author [Man22]. One major point of this paper is to extend the domain of validity of the $!$-functor formalism developed in [Man22] to allow certain ‘stacky’ maps. In the language of this extended formalism, we show that if G is a p-adic Lie group, the structure map of the classifying small v-stack $B\underline {G}$ is p-cohomologically smooth.

A closed Riemannian three-manifold $(Y,g)$ equipped with a torsion spin$^c$ structure determines a family of Dirac operators $\{D_B\}$ parametrized by a $b_1(Y)$-dimensional torus $\mathbb {T}_Y$. In this paper, we develop techniques to study how the topology of the locus $\mathsf {K}\subset \mathbb {T}_Y$ corresponding to operators with non-trivial kernel (the three-dimensional analogue of the theta divisor of a Riemann surface) depends on the geometry of the metric. As a concrete example of our methods, we show that for any metric on the three-torus $Y=T^3$ for which the spectral gap $\lambda _1^*$ on coexact $1$-forms is large, after a small perturbation of the family, the locus $\mathsf {K}$ is a two-sphere.

While the result only involves linear operators, its proof relies on the non-linear analysis of the Seiberg-Witten equations. It follows from a more general understanding of transversality in the context of the monopole Floer homology of a torsion spin$^c$ three-manifold $(Y,\mathfrak {s})$ with a large $\lambda _1^*$. When $b_1>0$, this gives rise to a very rich setup and we discuss a framework to describe explicitly in certain situations the Floer homology groups of $(Y,\mathfrak {s})$ in terms of the topology of the family of Dirac operators $\{D_B\}$.

The compactly supported $\mathbb {A}^1$-Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an analogue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties $\mathrm {K}_0(\mathrm {Var}_k)$ taking values in the Grothendieck-Witt ring $\mathrm {GW}(k)$ of the base field k. The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, the first author and Pál construct a power structure on $\mathrm {GW}(k)$ and show that the compactly supported $\mathbb {A}^1$-Euler characteristic respects these two power structures for $0$-dimensional varieties, or equivalently étale k-algebras. In this paper, we define the class $\mathrm {Sym}_k$ of symmetrisable varieties to be those varieties for which the compactly supported $\mathbb {A}^1$-Euler characteristic respects the power structures and study the algebraic properties of the subring $\mathrm {K}_0(\mathrm {Sym}_k)$ of symmetrisable varieties. We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported $\mathbb {A}^1$-Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.

This is the first part of a series of papers devoted to the study of linear cocycles over chaotic systems. In the present paper, we establish the existence of such cocycles that $\mathcal {C}^\alpha $-stably exhibit fiberwise bounded orbits ($\alpha>0$). The proof is based on a new mechanism which yields stable elliptic-type behavior in $\mathrm {GL}(d,\mathbb {R})$ or $\mathrm {SL}(d,\mathbb {R})$ cocycles. Moreover, we show that this phenomenon is $\mathcal {C}^0$-dense among $\mathrm {SL}(d,\mathbb {R})$ cocycles over a shift of finite type without dominated splitting.

The notion of strong 1-boundedness for finite von Neumann algebras was introduced in [Jun07b]. This framework provided a free probabilistic approach to study rigidity properties and classification of finite von Neumann algebras. In this paper, we prove that tracial von Neumann algebras with a finite Kazhdan set are strongly 1-bounded. This includes all property (T) von Neumann algebras with finite-dimensional center and group von Neumann algebras of property (T) groups. This result generalizes all the previous results in this direction due to Voiculescu, Ge, Ge-Shen, Connes-Shlyakhtenko, Jung-Shlyakhtenko, Jung and Shlyakhtenko. Our proofs are based on analysis of covering estimates of microstate spaces using an iteration technique in the spirit of Jung.