We prove that the conformal dimension of any metric space is at least one unless it is zero. This confirms a conjecture of J. T. Tyson [23, Conj. 1.2]

We prove that for any field $k$ of characteristic $p>0$, any separated scheme $X$ of finite type over $k$, and any overconvergent $F$-isocrystal $\mathcal{E}$ over $X$, the rigid cohomology $H_{\mathrm{rig}}^i(X,\mathcal{E})$ and rigid cohomology with compact supports $H_{c,\mathrm{rig}}^i(X,\mathcal{E})$ are finite-dimensional vector spaces over an appropriate $p$-adic field. We also establish Poincaré duality and the Künneth formula with coefficients. The arguments use a pushforward construction in relative dimension $1$, based on a relative version of Crew's [Cr] conjecture on the quasi-unipotence of certain $p$-adic differential equations

Eisenbud, Popescu, and Walter [4] have constructed certain special sextic hypersurfaces in ${\mathbb{P}}^5$ as Lagrangian degeneracy loci. We prove that the natural double cover of a generic Eisenbud-Popescu-Walter (EPW) sextic is a deformation of the Hilbert square of a $K3$-surface $(K3){}^{[2]}$ and that the family of such varieties is locally complete for deformations that keep the hyperplane class of type $(1,1)$; thus we get an example similar to that (discovered by Beauville and Donagi [2]) of the Fano variety of lines on a cubic $4$-fold. Conversely, suppose that $X$ is a numerical $(K3){}^{[2]}$, suppose that $H$ is an ample divisor on $X$ of square $2$ for Beauville's quadratic form, and suppose that the map $X⇢\left|H\right|{}^{\vee }$ is the composition of the quotient $X\to Y$ for an antisymplectic involution on $X$ followed by an immersion $Y\hookrightarrow \left|H\right|{}^{\vee }$; then $Y$ is an EPW sextic, and $X\to Y$ is the natural double cover

Fix an algebraic space $S$, and let $\mathcal{X}$ and $\mathcal{Y}$ be separated Artin stacks of finite presentation over $S$ with finite diagonals (over $S$). We define a stack ${\underline{\mathrm{Hom}}}_S(\mathcal{X},\mathcal{Y})$ classifying morphisms between $\mathcal{X}$ and $\mathcal{Y}$. Assume that $\mathcal{X}$ is proper and flat over $S$, and assume fppf locally on $S$ that there exists a finite finitely presented flat cover $Z\to \mathcal{X}$ with $Z$ an algebraic space. Then we show that ${\underline{\mathrm{Hom}}}_S(\mathcal{X},\mathcal{Y})$ is an Artin stack with quasi-compact and separated diagonal

A metric space $X$ has Markov-type $2$ if for any reversible finite-state Markov chain $\{Z_t\}$ (with $Z_0$ chosen according to the stationary distribution) and any map $f$ from the state space to $X$, the distance $D_t$ from $f(Z_0)$ to $f(Z_t)$ satisfies $\mathbb{E}(D_t^2)\le K^{2}t\mathbb{E}(D_1^2)$ for some $K=K(X)<\infty$. This notion is due to K.Ball [2], who showed its importance for the Lipschitz extension problem. Until now, however, only Hilbert space (and metric spaces that embed bi-Lipschitzly into it) was known to have Markov-type 2. We show that every Banach space with modulus of smoothness of power-type $2$ (in particular, $L_p$ for $p>2$) has Markov-type $2$; this proves a conjecture of Ball (see [2, Section 6]). We also show that trees, hyperbolic groups, and simply connected Riemannian manifolds of pinched negative curvature have Markov-type $2$. Our results are applied to settle several conjectures on Lipschitz extensions and embeddings. In particular, we answer a question posed by Johnson and Lindenstrauss in [28, Section 2] by showing that for $1<q<2<p<\infty$, any Lipschitz mapping from a subset of $L_p$ to $L_q$ has a Lipschitz extension defined on all of $L_p$