7 problems

Let $Y$ be a $CAT(0)$ space, let $u_0:\mathbb R^n\to Y$ satisfy $E(u_0)<\infty$ and $d(u_0,Q)\in L^\infty(\mathbb R^n)$ for some $Q\in Y$, and let $u$ be the suitable weak solution…

Let $X$ be a finite-dimensional Alexandrov space with curvature bounded from below, let $Y$ be a $CAT(0)$ space, and let $u:X\to Y$ be harmonic. Jost–Lin conjecture. The map $u$ sh…

Let $A_S$ be an Artin group of type $A_n$, let $\pi:\Delta_S\to\mathsf C_S$ be the folding map to the Coxeter complex, let $H$ be a wall in $\mathsf C_S$, and let $K_H$ be the maxi…

Conjecture de rigidité du rang. Alors soit $X$ est un espace symétrique de type non compact et de rang supérieur, soit $X$ est un immeuble affine de dimension $\b1geq2$, soit $G$ c…

Let $G$ be an swm-graph. A Boolean-gated set of $G$ is a nonempty vertex subset $X$ such that every common neighbor of any two distinct vertices of $X$ belongs to $X$, and such tha…

Rank Rigidity Conjecture. The following three conditions should be equivalent: $X$ admits a rank one axis; $\operatorname{diam}(\partial_T X)>\pi$; and the induced action on…

Let $D$ be a bounded contractible domain in a Euclidean space of arbitrary dimension, and consider reflecting Brownian motions in $D$ together with co-adapted couplings. A coupling…