18 problems

Optimal-constant conjecture. The optimal value of $p(m,n)$ in the $\widetilde{A}_2$ case is

Schwartz–Shemanske's conjecture. For all $n\geq 3$,

Let $S$ be a Riemann surface, let $\mathfrak{t}$ be a maximal split toral subalgebra of $\mathfrak{p}\subset\mathfrak{g}$, and let $W$ be the Weyl group acting on $\mathfrak{t}$. A…

Let $d,q\in\mathbb{N}$ with $d\geq 2$, and let $f:\mathbb{N}\cup\{0\}\to\mathbb{N}$. Let $X$ be covered by a $q$-thick affine building of dimension $d$, and let ${\mathcal{F}}$ be…

Positive-density patterns conjecture. Every positive-density subset of the vertex set of an irreducible affine building contains these arbitrarily large prescribed configurations w…

Let $n$ be a positive integer and let $a,b,c,d$ satisfy $1\leq a,b,c,d<n$, $a+b>n$, and $a+b+c+d=2n$. Let $F$ be the cluster-algebra function on…

Let $X$ be a building, viewed as the set of its vertices, and let … be the complex of normed vector spaces whose degree-$n$ part is spanned by alternating $(n+1)$-tuples of vertice…

Let $X$ be an affine building of type $\widetilde A_2$ that is not isomorphic to the building associated to $\mathrm{PGL}_3(D)$, where $D$ is a finite-dimensional division algebra…

Let $L_1,L_2,L_3,L_4$ be lattices in $\mathbb{C}((t))^n$, and let $i_1,i_2,i_3,i_4$ be nonnegative integers with sum $n$. Let $f^t_{a,b}(L,M)$ denote the binary tropical determinan…

Let $L_1,\ldots,L_k$ be lattices in $\mathbb{C}((t))^n$, and let $i_1,\ldots,i_k$ be nonnegative integers with sum $n$. Let $f^t_{i_1,\ldots,i_k}(L_1,\ldots,L_k)$ be the tropical d…

Let $L,M,N$ be lattices in $\mathbb{C}((t))^n$, and let $i,j,k$ be nonnegative integers with $i+j+k=n$. For lattices $L_1,\ldots,L_r$, define the tropical determinantal valuation…

Let $x_1,x_2,x_3$ be points in the affine Grassmannian, and let $\omega_i,\omega_j,\omega_k$ be fundamental weights of $SL_n$ with $i+j+k=n$. For points $p,q$ in the affine Grassma…

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

Affine-building vanishing conjecture.

Vanishing conjecture. Similar vanishing theorems should hold for large-scale homology of higher-dimensional products of trees and for thick affine buildings.

Existence conjecture. If $\nu$ is a complete valuation, then an $\mathbb{R}$-building $I_{K,\nu}(G)$ always exists.

Affine-building characterization conjecture. $\Gamma$ is “isomorphic” to the building at infinity of some thick affine building of type $\widetilde{\mathsf{G}}_2$ if and only if th…

Let $X$ be an affine building modeled over $\mathbb{A}=\mathbb{A}(\mathrm{R},\Lambda,T)$, where $T$ is a proper translation subgroup of $\Lambda^n$. Let $A$ be an apartment identif…