7 problems

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 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 $L,M,N$ 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 $A$.

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…