43 problems

Lee–Szczarba conjecture. For a prime $p$ and $n\geq 2$, the induced map

Let ${\mathcal G}_k$ be the Grassmannian of $k$-dimensional singular subspaces of a building of type $\textsf{X}_{n}=\textsf{C}_{n},\textsf{D}_{n}$, and let ${\mathfrak G}_k$ be it…

Characterization conjecture. When $J_+=\emptyset$, the map $j_\beta$ is characterized by property d), namely compatibility with the Lie algebra filtrations of $\mathfrak g$ and…

Unfoldability conjecture. A billiard triangle in $\Delta$ with sides $\overline{ox_0}$, $\overline{ox_k}$ and $x_0x_1\ldots x_{k-1}x_k$, and with the special vertices $0,x_0,x_k$,…

Finite-moment random-walk conjecture. The theorem asserting that there exists a regular vector $\lambda\in\mathfrak{a}^{++}$ such that $(Z_n(\omega)o)$ is $\lambda$-regular almost…

Let $\underline{G}$ be a reductive group with involution $\theta$, let $\underline{H}$ be the associated subgroup, and let $\chi$ be a quadratic character of $\underline{H}$ trivia…

Cocycle-expansion conjecture. For every such lattice $\Gamma$,

Low-symmetry Menger-boundary conjecture. Hyperbolic groups with Menger curve boundary whose standard $2$-complexes have far less symmetry are graphically discrete.

Let $G$ be a semisimple real Lie group of real rank $r$ with finite center and no compact factors, and let Hitchin rays be rays in the representation or Higgs-bundle parameter spac…

Let a building of type $\widetilde{A}_n$ be a simplicial complex built from Coxeter complexes of type $\widetilde{A}_n$, and let its 1-skeleton be the graph whose vertices and edge…

Marquis's conjecture. The group $G$ has a global fixed point in $X\cup\partial X$.

Let $\Gamma$ be a group with geometric dimension $\operatorname{gdim}(\Gamma)$, acting properly and cocompactly on a locally finite, thick building. Action dimension bound for loca…

Let $q$ be the parameter in the construction of $X^n_\varnothing$, let $0\leq i\leq n-1$, and let $\Delta$ act on $C^i(X^n_\varnothing)$. The positive eigenvalues of $\Delta$ are a…

Consider the reduction process for constructing the pre-building $B_{\phi}^{\rm pre}$ from an $SL(3)$ spectral curve and its initial scaffolding, which records which edges are fold…

Let $X$ be a complete simply connected Riemann surface, let $\Sigma=\{\phi_1,\phi_2,\phi_3\}$ be an $SL(3)$ spectral curve, and let $A=\mathbb{R}^2$ be the standard apartment. A…

Higher-rank alternating-product conjecture. There exist $w_1,\ldots,w_n\in W$ such that:

Let $B_{i,\infty}^\pm$ be the stabiliser of the corresponding end, with subgroups $B_W^\pm$, $U_i$, and $W^{even}$ as defined in the paper. Structure decomposition conjecture. For…

Let $A^7=g\cdot A^7$ for some $g\in B_{i,\infty}^7$, where $A^7$ is the standard apartment and $U_i$ is the corresponding unipotent subgroup. Transitivity conjecture. For each such…

Pre-building Finsler isometric-embedding conjecture. The factorization $f:B^{\rm pre}_{\phi}\rightarrow B$ is an isometric embedding for the Finsler distance.

Bounded-region finiteness conjecture. On bounded regions, the process stabilizes in finitely many steps.

No-BPS finite stabilization conjecture. The sequence $Z_i$ is well-defined until it reaches a construction with no more fourfold points or sixfold points with triple fold lines; th…

No-BPS determinacy conjecture. If there are no BPS states, then an indeterminate case does not occur.

Folding-in versality conjecture. Every such map $F\rightarrow B$ extends to a map $G\rightarrow B$, although the extension need not be unique.

Building realization conjecture. The topological space $F(p)$ has a natural structure of an $\mathbb{R}$-building modeled on the affine space $A$.

Topological structure conjecture. The topological space $F(p)$ is Hausdorff and is a CW-complex.